Is your TASCAM connected to the computer with a USB cable?
If so, it might be - if one out of two of the four wires in it were broken somewhere along the cable's length - that replacing the cable would cause the device to work with the computer.
With batteries in it, but not connected to the computer, can it turn On, or does it fail to?
And if it fails to turn On by itself even with good batteries, then were leaking old batteries ever in its battery compartment?
If so, cleaning the battery compartment terminals, with cotton (e.g. cotton swabs) moistened with rubbing alcohol, until the orange or brownish is completely gone from the battery terminals, would help.
Let me know what you think, and what the above produces or suggests. I am not there to see the device in person, so at this point I'm just offering whatever advice may help or might help.