The Corfu General Hospital (state-owned) has a website with contact info, directions, etc. on it here
http://www.corfuhospital.gr/info1.htm

It looks as if they have haemotological goings-on on Tuesday, Wednesday and Thursday mornings. Give them a call, and see if you have to get an appoinment or what. There appear to be satellite medical centres of the hospital in the north also
http://www.corfuhospital.gr/kentra1.htm
http://www.corfuhospital.gr/iatreia1.htm
http://www.corfuhospital.gr/markos1.htm

There are private doctors, but I don't know how quickly they will be able to get blood test results/how much it will cost.

There are microbiologist who do blood tests quickly, the one in Acharavi (20 mins from Sidari) at the Pump will take the blood and do the test there, it is private. Othewise I am sure a Sidar doctor will do it privately and get the results quickly.