ZOB Center Controller reported extreme weather and ZMP Center not having the frequency to open another sector led to an oversaturation of aircraft flying through the corridor and multiple airspace violations.

Narrative

The northwestern corner of our airspace there are 3 different centers that match up to each other. There was extreme precipitation on the western edge of our airspace. There were reroutes for ZZZ and ZZZ1 in effect and most of the aircraft that were transitioning east to west or west to east were flying through this corridor.Needless to say all the high sectors ended up being saturated with aircraft as time went by. Also ZMP could not open another sector to help because; come to find out they didn't have the frequency because it was out of service.As time went by; aircraft were deviating more and more; making an already complex situation more complex. Then ZMP ended up telling us they could not take any more aircraft from us; all while there were multiple aircraft approaching their boundary without a handoff being accepted. The rest of the aircraft we attempted to turn back to the south to avoid ZMP; but they refused to turn because of the weather all around them. We were trying to keep aircraft clear of ZMP; but because of all the factors; we were unable to; resulting in multiple airspace violations. We tried to work with ZMP as much as possible due to the fact we still were talking to multiple aircraft that were flying in their airspace.Recommendation: There [needs] to be a better plan when there is known outages and weather conditions.