Ok, as a long shot, does the jigsaw have a brand & model? Any chance of a manual?
There may be someone of ecperience here, but for the most part transformers look like transformers. There are a few different footprints, but they come in a variety of windings & having them made to specifics is not very difficult & cost-effective. So, could be a wide variety of specifications.
With a circuit, you might get lucky with a part number, or even work out the specs.
The board looks as if it has done the miles.
Hope you get some help.