If you reverse engineer it keep in mind that magnetic field strength is determined by ampere-turns. That is what you need to match. Not simply wire length and gauge. This becomes a problem when you replace larger diameter old cotton covered enamel wire with same-gauge but smaller diameter modern cross-link polymer insulated magnet wire. You end up with either more turns from the same length, or less resistance from the same number of turns because of less length needed with the smaller new wire.***
If you use a different gauge, look up it's resistance per foot (or meter) and calculate how many turns will give the equivalent ampere-turns. If it is a series wound motor, for a different gauge you MUST involve the field resistance in the calculations as a series resistance when figuring amperes through the armature to match ampere-turns.
Motor acceleration from stop to running speed is a very complicated integration of several non-linear functions, so I'm not going to spill the math here. Simply put, if you want the same start/run characteristics try to stay near the original gauge and number of turns.
That armature appears to be in excellent shape. If it has a fault it is likely easily found and repaired without a complete rewind. Have fun.
*** Here's two 'cheat' methods to get new, thinner insulated wire to end up wound tight with the same number of turns: 1) build up the core slots with fiberglass tape to increase the spool diameter. 2) wind dental floss with several of the turns to take up space. Once you find what works on the first run (a few tries) the rest will be easy.