Buy one of those cheap third hand gadgets with the two alligator clips and integral magnifier from harbor freight tools. Also get a bright led light on it’s own stand and position where needed. Then get yourself some cardas silver solder. Here comes the most important parts....you must use the correct solder iron tip on low power. Use a pointed tip rather than the larger tips shaped like a regular head screwdriver. You want to be able to control the heat, so as not to over heat the tiny wire.
Lastly, place the cartridge connector in the jaws of one of the alligator clips on the third hand gadget and position under magnifier, and direct your light source accordingly. Then place the tiny wire, that has been pre tinned with solder, to the the second alligator clip of third hand gadget. It will take some time to get it all positioned correctly. You then have your 2 free hands to hold the soldering iron and solder. Another tip is to not try and strip wire with a tool. Use your finger nails to pinch wire and gently remove some insulation. A second tip would be to pre solder the connector, whereas you apply a small amount of solder to the entrance of the connector and then just reheat it as you insert the wire..In fact, when soldering it is best to tin both pieces of whatever you are soldering as it promotes an easier and quicker process.
Here’s a link to that gadget...
https://www.harborfreight.com/helping-hands-60501.html?cid=paid_google|||60501&utm_source=google...