After two long months, we finally got an apartment! We are located on an island called Lidingö immediately east of Stockholm. The landlord actually said we could have the apartment a month ago, but it took another month for the rental association to approve us. We are living in what is called a "bostadsrätt", which is a building where people buy membership in the building association to give them the right to live there. Because you are buying a portion of the building, and not a particular apartment, you do not technically own the apartment you live in, which means the building association actually retains the rights to determine who can and can't live there.
To get to work from our new apartment, Kevin takes a small local train that goes around the south side of Lidingö to Stockholm, then takes the subway from there to where he works. There are a few restaurants and a hardware store within walking distance, and we can take the train or a bus to the next little village to go grocery shopping. It seems to be a great apartment in a good area, so it all turned out well in the end.