Reset SMC without power button
If you open the system and disconnect all power, eventually the SMC has to restart, but that's not necessarily the same as a signaled clear.

We would need to have the schematics of the SMC and the firmware to know for sure, but if you don't feel you want to repair the keyboard and can't otherwise use the Mac without an SMC reset, I would disconnect the battery for maybe 24 to 48 hours. You've already done the brief disconnect of power so this likely is a situation where you need to have the boards replaced or repaired.

Assuming there's no damage due to repairs, $300 for a mail in repair was an option until the 2013/2012 models went vintage.


https://support.apple.com/en-us/HT201624


If it's purchased in Turkey or California, you still have time for the Apple repair at the approximate cost above.