It looks like the three ends that should be joined together inside the motor have been separated and additional wires taken from them to the outside. That makes both ends of the three windings accessible for experimentation.
My interest is in getting the motor running again in its intended setup. So joining back together those wires, removing any superfluous extra length and putting it back together.
It is good to have a reason for something not working, as that gives hope that it can be fixed.
The easiest way to more speed us simply more volts, and a controller that can manage the increased voltage and ideally limit current to a reasonable value.
If you want to read more about messing with the windings, which I think is not a useful thing to do with this kind of motor, look up WYE DELTA and you will see diagrams like this:
View attachment 45424
Commonplace with the starting of AC induction motors before the introduction of variable speed drives.