For people who are interested, I will share the 16K internal upgrade with a 128K SRAM. I didn't have a 62256 SRAM in my parts bin, but I did have a 128K SRAM. I decided to perform the upgrade with that.
I compared both datasheets and it turns out the pinouts are almost the same, the 62128 has 32 pins, the 62256 has 28 pins. That requires a little modification to make them work in a ZX81.
The modification is quite simple:
Cut pins 1, 2, 32 and 31 so they cannot short out on the PCB.
Take some wire and solder pins 1 & 2 together, wrap the wire around and solder pins 32 & 31 together and connect them with pin 30. This will tie them to +5 volt, effectively pulling them high.
Bend up pins 3, 28, 25 & 23 (1, 26, 23 & 21 on a 62256)
Solder a wire from pin 3 to the cathode of D1
Solder a wire from pin 28 to the cathode of D5
Solder a wire from pin 25 to the cathode of D3
Solder a wire from pin 23 to the cathode of D2
This will give you 16K of internal RAM on your ZX81, I found the wiring directions here:
http://blog.tynemouthsoftware.co.uk/2017/10/zx81-internal-16k-ram-reversible-version.html
I recommend soldering a 10K Ohm resistor between pins 29 and 30 (27 and 28 on a 62256) to make high resolution stuff work.
That's it basically. More RAM is possible, but that requires additional logic.