Við létum rafvirkja setja nýjan rofa við hurðina.

This is a very common Icelandic pattern:

láta + someone in the accusative + infinitive

It means to have/get/let someone do something.

So:

• Við létum rafvirkja setja...
  = We had an electrician install...

Here:

• létum = we had / we got / we let
• rafvirkja = an electrician
• setja = install / put

So the sentence is not saying we installed the switch ourselves. It says we caused or arranged for the electrician to do it.

Létum is the past tense, 1st person plural form of láta.

The verb láta can mean different things depending on context, including:

• let / allow
• make / have someone do something
• sometimes even leave

In this sentence, it is the causative use:
Við létum rafvirkja setja... = We had an electrician install...

So here it is closest to English had or got, not just plain let.

Because rafvirkja is in the accusative case.

After láta in this causative construction, the person who does the action is usually put in the accusative:

• láta einhvern gera eitthvað
  = make / have someone do something

So:

• rafvirki = nominative
• rafvirkja = accusative

That is why the sentence has:

• Við létum rafvirkja setja...

not

• Við létum rafvirki setja...

Because after láta, Icelandic normally uses a bare infinitive.

So you get:

• láta einhvern setja
• not láta einhvern að setja

This is similar to English in phrases like:

• We made him go
• not We made him to go

So setja appears directly after rafvirkja with no að.

Because rofa is the direct object of setja, so it must be in the accusative.

The noun is:

• rofi = a switch

In the accusative singular, it becomes:

• rofa

The adjective must agree with the noun in gender, number, and case:

• nýr rofi = a new switch
• nýjan rofa = a new switch, as a direct object

So:

• setja nýjan rofa
  = install a new switch

Here við hurðina means by the door, next to the door, or near the door.

It does not usually mean that the switch is physically on the door. It means its location is by the door.

Also:

• við takes the accusative
• hurðin = the door
• hurðina = the door, accusative singular

So:

• við hurðina = by the door

Because hurðina is definite, while rafvirkja is indefinite.

• rafvirkja = an electrician
• hurðina = the door

This is very natural in context:

• the electrician is just some electrician, not one already identified
• the door is a specific door, probably the one in that room or house

So the sentence means something like:

• We had an electrician install a new switch by the door

not

• We had the electrician install a new switch by a door

In this sentence, it is much more naturally understood as:

• We had an electrician install...
• We got an electrician to install...

Using let in English would suggest mainly permission:

• We let an electrician install it

But with a professional like an electrician, the usual meaning is that we arranged for the work to be done.

So although láta can sometimes correspond to let, here had is the best match.

This is the most neutral and natural order for this sentence.

Icelandic does allow some word-order flexibility, especially to change focus or emphasis, but this version is the straightforward one:

• Við létum rafvirkja setja nýjan rofa við hurðina.

It follows a very common pattern:

• subject + láta + person + infinitive + object + location

So while other orders may be possible in context, this is the form a learner should recognize as the normal default.

A very useful pattern is:

einhver lét einhvern gera eitthvað

which means:

someone had/made/got someone to do something

Applied here:

• Við = we
• létum = had
• rafvirkja = an electrician
• setja = install
• nýjan rofa = a new switch
• við hurðina = by the door

So the whole structure is:

We had an electrician install a new switch by the door.

This is one of the most important things to notice in the sentence.