"I cleaned up my house yesterday."
But that's not the sentence that is there. We have to go with the sentence that is there that is most correct.
If I understood everything well so far:
- the shi in shi-mashita is for "doing/making" something that is a noun.
- so, souji is a noun (not "to clean" but "a cleanup")
- which brings us to "no", meaning "the cleanup of the house".
But I'm not sure "wa" would be really wrong there...
And certainly I'm lost about the suggested translation having "des". From what I learned so far, your sentence is exactly what I would use.
I interpreted 家の掃除をした as "I did the housecleaning."
My recollection is that you can't have two を in the same sentence. However, if you drop the を between 掃除をした, you might be able to say 家を掃除した = I cleaned the house yesterday.
Would 家は掃除をした mean "Speaking of my house, I cleaned (it) yesterday"?
I don't know for sure which, if any, of these is more common and natural. Often, there are many ways to say things that are not necessarily wrong, but don't sound natural to native speakers.
の shows a close interrelationship between the nouns 家 and 掃除. Other particles (like に, を, and は) would show direction or would set 家 off as the noun of the sentence and turn 掃除 into a separate verb. の shows that the nouns work together — either as a compound noun or as a descriptive noun phrase.
You can look at 家の as either possessive or adjectival, since sometimes の is used to make one noun modify another noun. Either way, 家の describes 掃除. In English, "house" has the same form as a noun and an adjective, so whichever way the Japanese is intended, it translates to essentially the same thing in English.
If you translate 家の掃除をした as possessive, it'd be "The house's cleaning [I] did" or "The cleaning of the house [I] did"; if you translate it as adjectival, it'd be "The housecleaning [I] did." This is quite different in nuance from, say, 家は掃除をした, which would translate as "On the topic of the house, the cleaning [I] did."