What a frustrating situation. You do have the right to working heat, air conditioning, and other essential things that make your home habitable.
If you were in no way the cause of the broken heat, air, or appliance(s) and the lease doesn’t require you to maintain them, you may be able to terminate your lease, withhold rent, and/or seek damages. However, you typically are required to give the landlord ample opportunity to fix the problem themselves first. Usually, you've got to provide written notice to the landlord detailing each of the problems and requesting that each problem be remedied. The letter must also give the landlord a deadline (often about 2 weeks after they receive the letter) to fix or make their best effort to fix the issues. If the landlord refuses to comply with your demand, then you are at a point where you can choose which remedy (terminating, withholding rent, etc.) is best for your situation.
Also, a landlord would not generally be able to just plug something into your house, much less something that uses that much electricity. However, if the landlord properly notified you that he would be using your house and gave you a reduction on your rent, for instance, he may be able to. There is also a (fairly small) chance your lease allows for it. Your lease may also have other information about your available remedies and obligations (e.g. for the notice mentioned).
Ultimately, you may be able to legally terminate the lease, but need to comply with the lease and the law. I suggest you consult with a local attorney or, if income is an issue, consider contacting a legal aid center in your area.
The information and opinions I provide are based upon the limited facts provided. Each case, judge, and jury are different and my response is not intended to be legal advice. Laws vary from state to state - for advice on your specific circumstances, consult with an attorney or legal aid center in your area.