We're going to need all the details here to get this right, but a few things I'm going to note even before getting into the complete answer.
* Using a Mac for troubleshooting is ideal here. Windows won't have the right software natively to format or read this drive for Mac use. There are ways to do this, but it will be much more relevant to use it with the Mac for now. If you open disk utility, does it see the device?
* Drives from Macs made during this time frame are a bit hodge-podge and compatibility is not always intuitive. Many external enclosures specifically indicate they are not compatible with Toshiba drives. For example, previous iterations of OWC's Envoy Pro specify in the fine print it will not work with Toshiba-based Apple drives. Be sure to read everything to make sure your drive works with all parts.
* You also mentioned a new logic board. Is the logic board you installed the same model as the previous one? Some Airs have shells that will fit logic boards from more than one generation. My other thought is that the board may have a version of the firmware that is too old to use the replacement drive. But I would want to cover the other 2 points before getting here.