Not realising that I had a life time warrenty with my SD card, I shoved a very small amount of something called Polymoprh into the tab gap.
This allowed the card to become read/write again, although now it's stuck as read/write and I can't change that.
I'm guessing that a little Bluetak or something like that would do the same thing.
Otherwise check with the manufactory, they might be simply swap it out for you.
Assuming you can still read off the card, back it all up and just put it back on a new one if/when the manufactory sends a new card out.
If am not mistaken, taping the hole left by the broken piece of plastic will make it write protected again, just like we used to do to record on those old Betamax video tapes (hundred of years ago )
Just try putting a piece of tape over the whole slot where the slider would have been, or if that doesn't work too well (the reader ends up cutting it, etc), you could try a little bit of glue.