Can I break my rental agreement and get a full refund of my deposit if my landlord won't fix something?

Well, first thing is the direct answer to whether you can just break your lease and demand your full deposit – no.  However, that’s not the end of things.  And, unfortunately, the answer will be a little vague; that is, it comes down to your lease.  If the washer was part of the rental and your landlord agreed to fix things, then you may be able to get it fixed with a 7 day notice to fix.  Also, you could (stressing could) have an argument that without the fix the place is an unsafe place to live.  So, here’s what I would say you do – make an appointment with a landlord-tenant attorney and bring your lease.  Discuss everything that’s happened in a confidential environment.  Many attorneys will offer a half hour to hour long free consultation.  The attorney can give you your legal standing and the best options available to you.  Finally, and this is really a practical thing, do not use an extension cord to operate the washer… that is beyond extremely dangerous.