Unfortunately, your Lr catalogue is stored on the original Mac, so just logging into your account would not work. As a Windows user in your situation, and if the need for speed was essential, I would remove the hard drive from the malfunctioning PCI and connect it to a working example and try to access the catalogue from an installation of Lr on that one. Is that at all possible with your Mac setup?
Edit: do you have separate backups of your catalogue, in the cloud for instance, but can the catalogue held there be installed on a different Mac? I 've no idea if this is possible, but if you're a professional facing a deadline etc, it might be worthwhile investigating this possibility if it's impossible to move the hard drive as I suggested above.