Learn Hardcore German: Unsere Waschmaschine ist gestern kaputtgegangen, aber wir haben sie heute vom Techniker repariert bekommen. - Our washing machine broke yesterday, but today we got it repaired by the technician.

Verbs that show a change of state or movement, and that do not have a direct object, usually take sein.
Kaputtgehen means to break / to stop working, and it describes a change of state of the subject (die Waschmaschine).

Because the washing machine changes from working to broken, the verb behaves like other change-of-state verbs (e.g. einschlafen – ist eingeschlafen, sterben – ist gestorben), so the correct form is ist kaputtgegangen, not hat kaputtgegangen.

What exactly is kaputtgegangen? Is kaputtgehen one verb?

Yes. Kaputtgegangen is the past participle of the separable verb kaputtgehen.

Kaputt is originally an adjective (broken).
In kaputtgehen, it has become a separable prefix attached to gehen (to go), and together they mean to break / to stop working.

Forms:

Infinitive: kaputtgehen
Present: Die Waschmaschine geht kaputt.
Perfect: Die Waschmaschine ist kaputtgegangen.

So kaputtgegangen is one word because it is the participle of the single lexical verb kaputtgehen.

Why is kaputtgegangen written as one word and not kaputt gegangen?

For separable-prefix verbs in the perfect tense, German spelling rules say:

prefix + ge + verb stem + (en/t) → written as one word.

So:

kaputtgehen → kaputtgegangen
aufmachen → aufgemacht
anrufen → angerufen

Writing kaputt gegangen would suggest kaputt is just an independent adjective and gegangen is the main verb (went broken), but in standard usage kaputtgehen is treated as a fixed verb, so kaputtgegangen is written together.

Why is it unsere Waschmaschine and not unser Waschmaschine?

Waschmaschine is grammatically feminine in German: die Waschmaschine.

The possessive unser- must agree with the gender and case of the noun:

Nominative feminine singular: unsere
Nominative masculine singular: unser
Nominative neuter singular: unser
Nominative plural: unsere

In the sentence, unsere Waschmaschine is the subject (nominative, feminine singular), so the correct form is unsere, not unser.

What case is unsere Waschmaschine in, and how do you know?

Unsere Waschmaschine is in the nominative case.

You can tell because:

It is the subject of the sentence: it is the thing that breaks (it goes broken).
The verb kaputtgehen does not take a direct object; it only needs a subject.

So the structure is:

Wer/Was ist kaputtgegangen? → Unsere Waschmaschine.
(Who/What broke down? → Our washing machine.)

That Wer/Was? question asks for the nominative case.

What does sie refer to in wir haben sie heute vom Techniker repariert bekommen, and why not es?

Sie refers back to unsere Waschmaschine.

In German, pronouns follow grammatical gender, not natural gender or the English idea of it:

die Waschmaschine (feminine) → sie (singular feminine pronoun)
der Kühlschrank (masculine) → er
das Auto (neuter) → es

So you must say:

Wir haben sie repariert bekommen. (about die Waschmaschine) not
Wir haben es repariert bekommen.

Using es would be grammatically wrong here, because es corresponds to neuter nouns (das-words), not to feminine ones.

Why is the object pronoun sie placed before heute and vom Techniker?

German has a fairly regular order for elements in the middle of a clause:

Conjugated verb in 2nd position (here: haben).
Pronoun objects.
Time, manner, place, other details.
Non-finite verbs (participles, infinitives) at the end.

In wir haben sie heute vom Techniker repariert bekommen:

wir – subject
haben – conjugated verb (position 2)
sie – object pronoun (comes early, right after the finite verb)
heute – time
vom Techniker – who did it (prepositional phrase)
repariert bekommen – verbal cluster at the end

So sie comes before heute because short object pronouns tend to stand as early as possible after the finite verb.

What does the construction repariert bekommen mean?

Repariert bekommen is a very common German construction meaning to have something repaired (by someone) or to get something repaired.

Pattern:

jemand bekommt etwas gemacht
→ someone has something done / gets something done.

In your sentence:

wir – the people affected
sie – the thing (the washing machine)
repariert – what was done to it
bekommen – marks this get/have something done meaning

So wir haben sie vom Techniker repariert bekommen =
We had it repaired by the technician / We got it repaired by the technician.

How is repariert bekommen different from a normal passive like sie ist repariert worden?

All three are possible, but they focus on different things:

Wir haben sie vom Techniker repariert bekommen.
- Focus: we arranged it / we are affected by it.
- Very natural in conversation; sounds like we got it repaired.
Sie ist (vom Techniker) repariert worden.
- Passive with werden.
- Focus: the action on the washing machine; the doer (Techniker) is optional.
- Closer to English It was repaired (by the technician).
Wir haben sie vom Techniker reparieren lassen.
- Causative with lassen.
- Focus: we caused someone to repair it.
- Roughly: We had the technician repair it.

In everyday speech, repariert bekommen is very common when you want to say that something for you got done by someone else.

Why is it vom Techniker and what does vom mean?

Vom is the contraction of von dem:

von
- dem → vom

Von always takes the dative case, so:

der Techniker (nominative, masculine)
dem Techniker (dative, masculine)

Thus vom Techniker literally means from the technician, but in this context it corresponds to English by the technician (the person who did the repair).

You could also say:

von dem Techniker – more explicit, slightly more emphatic or formal. But in normal speech, vom Techniker is more natural.

Why is the perfect tense used (ist kaputtgegangen, haben ... bekommen) instead of a simple past form like ging kaputt, bekamen?

In spoken German, the Perfekt (present perfect) is usually used to talk about past events, especially in everyday conversation.

Unsere Waschmaschine ging gestern kaputt and
Unsere Waschmaschine ist gestern kaputtgegangen
mean essentially the same, but the second is more typical in speech.
Wir bekamen sie heute repariert is grammatically possible, but sounds rather formal or literary;
Wir haben sie heute repariert bekommen is what you would hear in normal conversation.

So the sentence uses the perfect because it is the default past tense in spoken German in most regions.

Why does the sentence use aber and not sondern?

Both aber and sondern can be translated as but, but they are used in different situations:

aber introduces a simple contrast or addition:
- Es ist teuer, aber gut. – It is expensive but good.
sondern is used only after a negation to correct or replace what comes before:
- Es ist nicht teuer, sondern billig. – It is not expensive, but (rather) cheap.

In your sentence there is no negation before the conjunction:

Unsere Waschmaschine ist gestern kaputtgegangen, aber wir haben sie heute ... repariert bekommen.

So you must use aber, not sondern.

Could you also say Unsere Waschmaschine war gestern kaputt, and what is the difference from ist gestern kaputtgegangen?

Yes, you can say Unsere Waschmaschine war gestern kaputt., but it means something slightly different:

Unsere Waschmaschine ist gestern kaputtgegangen.
Emphasizes the event of breaking down that happened yesterday (the moment of change).
Unsere Waschmaschine war gestern kaputt.
Emphasizes the state of being broken during yesterday (it was in a broken condition yesterday; maybe it broke even earlier).

In context with aber wir haben sie heute ... repariert bekommen, your original version with ist gestern kaputtgegangen nicely contrasts:

Yesterday: the breaking happened.
Today: the repair happened.