There is quite a long thread on here about SGK, which might help you, but as far as I can gather .....
Your wife will have to have a medical before she can now join in her own right. Apparently there is a notice on the SGK office door, in English, to this effect. I read on here that you get a form and go to the Devlet Hospital for the health check and I think the cost is about 500TL.
You will be kept on your existing membership, you can check on this site:
https://esgm.sgk.gov.tr/Esgm/ to see if you are registered.
You will both have to pay separate premiums, i.e. double, if your wife is accepted.
You probably will need a translator or at least a Turkish speaking friend to help you.