Janmack - If you have or have ever had a Resident's Permit you can get your Yabanci Kimlik number, whereas if you have only had the 90 day tourist visa's you can't. The reason being that you need an RP number to enter on the Kimlik application.
Here is a link to the site to get your Kimlik number and you can print it off and keep it with your RP.
http://tckimlik.nvi.gov.tr/Web/ForeignQueryIdentityNumber.aspxAnd to stay on topic, I'm just as confused as everybody else as to whether this insurance is going to be compulsory for all resident permit holders, pensioners as well (if not privately insured) or just people of working age. I await more info with baited breath - the "For Sale" sign is already up !!!