There is actually a sale from Europe till 12.Jun.
For example LHR-TXL or DUS for GBP 89 in ecco and than TXL-LHR-MIA // MIA-JFK// LHR-TXL for a total off 1879 Euro ( 1598 GBP)
or much better from Italy (Rome or Malpensa ) LHR- FCO or MXP for 149 GBP in Ecco and then FCO- ORD- MIA or FCO-JFK-MIA and back MIA-LHR-FCO or MXP for 1700 Euro ( 1450 BGP ) … with an option for a cheep Firstclass travel one way of additional 1000 euro ( total 2700 Euro = 2200 GBP )
for the Return-flight check your bags to London only
btw why not stay for one night in Berlin or in Rome??