Back to the maker and ask for the service manual. As to the missing solder points why not fix those up? You are talking first year tech work here.
Also, as a workaround till then, try USB keyboards.
I have a Toshiba Satellite L750 laptop, model number PSK2YA-0DN02S, as per the laptop case model number and confirmed with Belarc Advisor. Recently, I replaced my failing original motherboard (A000081570-DABLDDMB8D0 REV D) with a second-hand motherboard (A000080820-DABLBDMB8E0 REV E). Long story but idiot merchant sent incorrect motherboard....
When I check Belarc Advisor, it states that the laptop system model number is now PSK2YQ-06Y020, reflecting motherboard change. I think this motherboard is from a European sold laptop where as my original laptop was purchased in Australia.
This motherboard appears to be functioning correctly in my laptop, a little slower BUT the original keyboard will not work with the replacement motherboard.
I have also noted that the original motherboard keyboard connector has some of it's contacts not soldered whereas all the contacts are soldered on the replacement for motherboard. The origial keyboard has not been damage and previously functioned correctly. The original laptop keyboard part number is MP-09N13US-698.
Anyway, can anyone please advise me of the CORRECT keyboard part number to order for the replacement for motherboard (A000080820-DABLBDMB8E0 REV E), model number PSK2YQ-06Y020.