The recipe calls for 350 degrees for 10 minutes, yet my oven only goes up to 260 degrees. How long should the cake stay in the oven?
Convert 350°F to 175°C - I don't think there's any lower temperature issue here.
If your oven goes up to 260 degrees, I'm guessing that's 260°C, not 260°F. (260°C is 500°F, so that's not a surprising maximum temperature to see, whereas a max of 260°F would be a pretty useless oven.)
A cake recipe that calls for 350 degrees is most likely 350°F, especially if you happen to know it's an American recipe. 350°F is an extremely common temperature in American recipes, possibly the most common. (A baking temperature of 350°C is really high and pretty unlikely - pretty much just for things like pizza, and certainly not cake.)