I do not believe you can power it up using the keyboard, but the power button itself can be replaced fairly inexpensively.
It looks like from the parts list, that their were two types of power boards available, one for WiFi models (Part No. 5C50N91232) and another for LTE (cellular) models (Part No. 5C50N38127).
Replacing the part requires prying open the case using a large guitar pick or other non metallic (conductive) pry tool, disconnecting the battery (very important), replacing the power board, and then reassembling (make sure to reconnect the battery).
If you are not comfortable doing the job yourself, any good computer/phone repair shop should be able to perform the job while you wait.
MllX 320-10lCR 80XF Parts List
Note: The parts listed below were found searching for the part numbers here in the USA, but you might try searching in your region to see if you can find one closer to reduce shipping costs etc.
Power Board(WIFI) B 80XF Part: 5C50N91232
Power Board(LTE) B 80XF Part: 5C50N38127
Lenovo Miix 320 Disassembly