Baking at a lower temperature than the recipe calls for 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.)