JFK-HEL-KRK is the fastest, though if the dayflight is an option, then JFK-LHR dayflight, overnight, LGW-KRK would get you into KRK by late morning and probably more refreshed than redeying to HEL.

Alternatively, LO flies JFK-KRK nonstop on Mondays, and you can start your RTW from KRK. Oneway C fares on LO are pretty cheap, I see the August Mondays for $1830 a/i. You could then put off flying the last few coupons to KRK for a year; if you fly to LHR or HEL a lot, you could just throw away the KRK part and get a useful oneway out of it.