If I leave my 12-month lease early, what steps does the landlord need to take to mitigate the damages?

Hopefully you have a good reason to break the lease and are on good terms with the owner. A frank and friendly chat with the owner and a polite request to work with you to terminate the lease agreement and move out on a specific date may save you headaches in the future. If there are two or more of you renting, go together to speak with the owner. Two memories are better than one. Maybe, if rents are higher now than when you moved in, you can use that as a selling point with the owner--it's better for the owner to get someone else in there now. Request that he / she start advertising the place for rent again and see what he / she says. Take notes of what you discuss and agree upon, and then write him / her a polite and respectful letter recapitulating what was said and agreed. If nothing else, make sure to explain in your letter why you need to break the lease and ask the owner to try to find a replacement lessee as soon as possible to ease the burden on you both. This letter may end up as valuable evidence in the event of litigation later, so make it nice and comprehensive, and of course keep a copy. Make sure to leave the place in great condition, and have a walk-through inspection with the owner before you leave. Take pictures or video to prove that you did not damage anything. If the owner sues you for something, you might have "failure to mitigate damages" as an affirmative defense, and it will be up to the trier of fact to decide whether the owner did or did not mitigate damages, depending on the evidence. If the judge / jury finds you more believable and sympathetic than the owner, you have a much better chance of a good outcome.

The landlord needs to take reasonable steps to try and rent the unit. But, you will be on the hook for the rent for the balance of your lease until it does get leased. Reasonable steps might include putting ads in local paper or publication, listing it with a real estate agent, or anything he might have done before he rented it to you. If he does those things, and still can't find a suitable tenant, then he could seek the balance of rent from you. As for when you should leave, you can leave any time you want. The longer you stay, the less rent you will be responsible for. Maybe you should try and find a replacement tenant before you leave.