Learn Icelandic: Síminn var bilaður hvort eð er, þannig að þessi klaufaskapur skiptir ekki máli. - The phone was broken anyway, so this clumsiness doesn't matter.

It translates directly to "anyway" or "in any case." As the lesson notes, it is an invariable phrase, meaning it never changes its form to match gender, number, or case. Notice that it sits at the very end of the first clause, which is the exact same place you would naturally put "anyway" in English.

Why does the word for "broken" end in -ur (bilaður)?

Here, bilaður acts as an adjective describing síminn (the phone). Since sími is a masculine noun in the nominative case, the adjective must agree with it. If we were talking about a neuter word like tækið (the device), it would be tækið var bilað.

Is þannig að the most natural way to say "so" in this context?

Yes. While þannig normally means "thus" or "in that way," adding að turns it into a conjunction meaning "so that" or "therefore." It is perfectly suited for linking a cause (the phone was already broken) to an effect (the clumsiness does not matter).

What causes the word mál to become máli at the end of the sentence?