It sounds like the minute switch has gone bad, or possibly the solder connections on the circuit board are broken. Unfortunately, it's hard to find anyone to work on a small appliance like this because of the cost. Most service shops don't take this kind of item for repair because it will probably cost more to repair than to replace. You may know someone with soldering skills who likes to tinker, and would be willing to open it up to check those connections. If the switch is bad, it's probably
this style. You can see how long the shaft is and see if one is available close to that length. The switch is cheap, but shipping will add a few dollars. Still, if you are willing and able to do it yourself you should be able to get this working. Good luck!