Solange der Waschsalon um die Ecke offen ist, brauchen wir nicht sofort eine eigene Waschmaschine.

Because solange is a subordinating conjunction. In German, a subordinating conjunction sends the conjugated verb to the end of its clause.

So:

• der Waschsalon ... ist = the laundromat ... is
• with solange: Solange der Waschsalon ... offen ist

This is very common with words like weil, dass, wenn, obwohl, and solange.

German main clauses follow the V2 rule: the conjugated verb must come in the second position.

Here, the entire subordinate clause comes first:

• Solange der Waschsalon um die Ecke offen ist, = position 1

So the verb of the main clause must come next:

• brauchen = position 2
• wir comes after it

That is why you get:

• Solange ..., brauchen wir ...

not

• Solange ..., wir brauchen ...

Here solange means as long as.

It introduces a condition that remains true for a period of time:

• Solange der Waschsalon ... offen ist = As long as the laundromat ... is open

It can sometimes also feel close to while, depending on context, but in this sentence as long as is the most natural match.

Also, this solange is written as one word. That is the normal spelling when it is a conjunction.

Because der Waschsalon is the subject of the first clause.

In:

• der Waschsalon ... ist offen

the laundromat is the thing that is open, so it is in the nominative case.

• der Waschsalon = nominative masculine singular
• den Waschsalon would be accusative masculine singular, which would be used for a direct object, not the subject

um die Ecke is a fixed expression meaning around the corner.

So:

• der Waschsalon um die Ecke = the laundromat around the corner

Why die Ecke?

• Ecke is a feminine noun: die Ecke
• the preposition um takes the accusative here
• feminine accusative singular is still die

So:

• um die Ecke

Even if you do not analyze it every time, it is worth learning as a chunk: um die Ecke.

Because offen is a predicate adjective here, not an adjective directly in front of a noun.

Compare:

• der offene Waschsalon = the open laundromat
  Here the adjective comes before the noun, so it takes an ending.

But in your sentence:

• der Waschsalon ist offen = the laundromat is open
  Here offen comes after sein and describes the subject, so it does not take an ending.

This is just like:

• Das Haus ist groß
• Die Tür ist offen
• Der Kaffee ist heiß

Because nicht is negating sofort here.

The idea is not that we do not need a washing machine at all, but that we do not need one immediately.

So:

• brauchen wir nicht sofort eine eigene Waschmaschine

means roughly:

• we do not need our own washing machine right away

If German wanted to negate the noun more directly, you would expect something different, such as using keine in a different sentence. But here the sentence is really about the timing: not immediately.

eigene means own.

So:

• eine eigene Waschmaschine = our own washing machine

It adds the idea of possession and contrast:

• not a public one
• not someone else’s
• not a laundromat’s machine
• but one belonging to us

The adjective ending -e appears because:

• Waschmaschine is feminine
• it is the direct object of brauchen, so it is in the accusative
• with feminine singular eine, the adjective takes -e in this pattern

So:

• eine eigene Waschmaschine

Because it is the direct object of brauchen.

In:

• wir brauchen eine eigene Waschmaschine

the subject is:

• wir

and the thing needed is:

• eine eigene Waschmaschine

German marks direct objects with the accusative case. For feminine singular, the article eine looks the same in nominative and accusative, but it is still accusative by function.

Yes. In German, a subordinate clause is normally separated from the main clause by a comma.

So this is correct:

• Solange der Waschsalon um die Ecke offen ist, brauchen wir nicht sofort eine eigene Waschmaschine.

That comma is not optional in standard German.

Yes. German allows both orders.

You could also say:

• Wir brauchen nicht sofort eine eigene Waschmaschine, solange der Waschsalon um die Ecke offen ist.

The meaning stays basically the same.

The main difference is emphasis:

• starting with Solange ... highlights the condition first
• starting with Wir brauchen ... highlights the main statement first

Both are natural German.

Here brauchen simply means to need and takes a noun as its object:

• eine eigene Waschmaschine brauchen = to need a washing machine

German brauchen can also appear with an infinitive in some contexts, especially with negation:

• Wir brauchen nicht zu warten. = We don’t need to wait.

But in your sentence it is the straightforward noun-object use:

• Wir brauchen ... eine Waschmaschine.