First step, talk to the authorities that issued your JAA and find out what they exactly want. Unfortunately, not every country treating it the same way.
I went through this last year holding a JAA (never used it) ready to expire. To convert it to EASA, the authorities required a proficiency check. Since all my ratings were expired, it was a requirement to get a minimum of three hours of training to get re-typed on an expired rating. The cheapest route for me was to get typed in the Cabri G2. Training and check in three hours. Was well worth the money.