Learn Russian: У нас есть дешёвое жильё на два дня. - We have cheap accommodation for two days.

When expressing the intended duration of a result (like securing accommodation for a planned period of two days), Russian uses на plus the Accusative case. If you used для, it would mean 'intended for the benefit of', which doesn't fit a time period.

If на triggers the Accusative case, why is дня in the Genitive case?

The preposition на does trigger the Accusative case for the whole phrase, which dictates the form of the number: два is in the inanimate Accusative (which looks identical to the Nominative). However, the standard rule for numbers still applies internally: after два, три, or четыре in the Nominative or inanimate Accusative, the following noun must be in the Genitive singular. So, день becomes дня.

What gender is the word жильё and what case is it in here?

жильё is a neuter noun, and here it is in the Nominative case. In Russian 'to have' constructions (у нас есть), the thing you possess is actually the grammatical subject of the sentence, literally translating to 'By us exists cheap accommodation.'

Why does the adjective дешёвое end in -ое if the noun жильё ends in a soft vowel?