I want to buy the house in Windhelm but the Steward just says that there was some unpleasantness recently and that its not available. Did I break something by completing the Imperial Legion quest chain?

Windhelm is the Stormcloaks city, so maybe it's glitched since you sided with the Imperials

The Empire put a different person in the Jarl's seat and he told me that I could buy a house in the city. Guess he was just kidding.

The unpleasantness is that there is a murderer stalking women in the city. You need to deal with that and help the general populace 3 (or is it 5) times and you can get the house. The quest you need to begin is called Blood on the Ice. Fair warning.....it's bugged for a lot of folks. You may not be able to start it.

If the quest doesn't pop up try leaving the city for a few days and coming back. You might also head into the inn directly in front of you as you enter the city and ask about rumors. Just keep asking until they mention someone or something killing women in town recently. There should be a fresh dead body in the graveyard with a guard and townsfolk around. That should be the start of the quest if I remember correctly.

Very bugged quest to get the house. I ended up getting stuck with a bug and couldn't get it. Read the bug reports carefully if you want any chance of getting it.

I actually got it and upgraded it, but there was an essentual enemy living in it also that I had to beat down every time I entered in the house. I finally just gave up once I realized I couldn't get past the part in the quests where I was supposed to kill him.

You need to help five residents of the hold and Blood on the Ice (which counts towards helping residents). Unfortunately Blood on the Ice is ridiculously buggy and easy to break. I would recommend checking out the wiki for that quest and following the instructions exactly if you want that house.

I've already helped 5 people in the town according to the quest counter. The Innkeeper doesn't have an option to ask about rumors, nor does anyone else in there. There was a dark elf in the Gnisis Corner Club that mentioned them but it didn't start a quest. And the Talk page for this quest on the unofficial elder scrolls pages site is overflowing with people trying to figure out how to trigger this quest. Seems like a significant oversight to only allow the event to trigger one time, and then stash the event away in a place where the player may not have any reason to go.