You've put yourself in a tough situation. Hindsight being 20/20, here's two things you should have done: 1) do your Cap-X immediately after closing. 2) Put your team together before, or as soon after buying properties as possible. The team clearly needed to include a HVAC company that could respond quickly.
If your electrical system can handle it, buy more electric heaters. Do some research and only get the safest ones, least likely to start a fire. Get enough of them so the tenant can adequately heat the apartment.
Talk to your insurance agent about what, if anything, your insurance will cover for temporary lodging for them.
Call all the local HVAC supply houses until you find one that has a suitable furnace in stock rather than wait for one that won't arrive for two weeks. With enough phone calls, you might be able to find a supplier/contractor who can respond faster than two weeks.
You may have to lower the rent for a month and/or help cover lodging.
Clearly communicate all that you are doing, and what the problems are to your tenant so that they don't thing you aren't doing anything and just don't care.