I've completed the Battle for Falkreath for the Stormcloaks, and I was told to purchase the property from the Steward.

When I went to the Steward, he said that the home wasn't available at the time, due to "unpleasantness."

I assume he's talking about the Blood on the Ice quest. However, that quest has been bugged for me, and I can't start the quest.

What do I do? I want to buy every house in the game, so I need to know how to get the home.

If the quest is bugged then it is unlikely that you would be unable to purchase it because that quest needs to be completed to allow it.

I had understood that you can buy the house, and that the quest begins as soon as you enter it.

I would say 'console', if you're playing on PC, that is. Otherwise, earlier save?