Rice cooker cake!
There's a cake recipe there that I tried and didn't really love, but I think the basic theory would work with pretty much any cake recipe you already have. Just mix the batter up, pour it into the rice cooker, and set it to "warm" for about 40 minutes. The slow low-heat cooking will give it a slightly spongy texture and probably works best with a batter that's fairly liquid. If you're impatient you can try other settings, but a fuzzy logic cooker like the one I have probably won't have much of a clue about how long to bake your cake for.
Another option is to steam the cake. Put two cups of water in the rice cooker and set it to "steam" or "cook" for a few minutes. Meanwhile, pour the cake batter into a pan that fits inside the cooker (or a few cups/ramekins if you have a small rice cooker like I do) and loosely cover the pan with foil. Carefully set it into the hot water bath. Then use the "cook" or "steam" setting to steam the cake(s) until done. This non-vegan recipe and this non-vegan recipe use that method.
Even after the oven is fixed, I plan to experiment further with this! I think it would be great for people in small apartments/dorms/barracks who don't have access to a full-size stove.