I'm not seeing much availability out of either JFK or ORD, except for Emirates at 138k pts + $1,244 in taxes in biz class. There is also the Qatar seats at 175k pts, but that's very steep (when the calendar opens it's significantly less).
Until January 6th Chase has a tiered transfer promo (depending on the size of the transfer, from 15 to 25% bonus points).
If you can stomach economy class then the below direct flight would probably be your best option (AP is a transfer partner of Chase). You can see there is availability on June 27-30th.
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This option out of LHR on June 28th is interesting:
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There is the below option from Heathrow on June 28th for 115k points. Saves points, but you still have to get to London. On the plus side there are several points hotels close to LHR that you can overnight at. If it's feasible for, it might be a good option.
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Coming back this is the best option that I see into JFK on July 12th:
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And the below on July 10th. I've flown Swiss Air in biz and first class and it's a very nice hard and soft product. Have never flown Ethopian.
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There is also this option:
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