First off - welcome to the forum!
It should be mentioned that if you don't have the legal right to live and work in the US (e.g greencard; US-Passport) it's pretty much pointless to do the JAA (CPL) to FAA conversion.
Basically there is no such thing like a license conversion/validation beyond the PPL level, no matter if it's FAA to JAA or vice versa!
No matter what level of license you hold, the FAA will only issue a PPL license which is based and valid in conjunction with a current foreign license only. For further details see link below:
Electronic Code of Federal Regulations:
For the issuance of a full independent FAA PPL check the following link:
Electronic Code of Federal Regulations:
As you already said correctly, in order to get an FAA CPL/IR you would need to pass the respective theoretical and practical tests.
FAA instrument rating requirements (Part 61.65)
Electronic Code of Federal Regulations:
FAA CPL requirements (Part 61.121)
Electronic Code of Federal Regulations:
Since you already hold the JAA CPL/IR you probably comply with (most) of the aeronautical experience required for the FAA CPL/IR and only little or no additional flight instruction would be needed to be ready for the check-ride.
Under Part 61 (Part 141 doesn't apply to you, it's only relevant for ab-initio candidates) no classroom instruction is required to prepare for the knowledge/theoretical test. Preparation for the written test can be done all by yourself at home. There are several commercial vendors providing home study "courses"/material which are approved by the FAA (e.g. like
ASA: FAA Test Preparation) you can find it on Amazon as well.
It's also possible to take the FAA knowledge tests in Europe, you'd only need an endorsement form a FAA instructor (which can be obtained from me, just drop me a pm).
For further information on FAA testing in Europe see the following link:
http://www.pprune.org/professional-p...st-europe.html
Regarding the "conversion" I'd recommend to get in touch with a flight school in the US and see what they say would be the best approach to get the "conversion" done (e.g.
License Conversions: ICAO to FAA).