I've never been able to sleep on an overseas flight. Some people can some people can't. I also don't care to try "sleep aids" - some people can sleep on a plane just fine without them, apparently.
My strategy for combating jet lag is to 1) rest up AS MUCH AS POSSIBLE the few nights before the flight. The worst jet lag I've ever had was at the start of a trip where I was stressed before leaving for Europe and didn't sleep well the few nights before. First night in Italy, I slept like a baby, about nine hours. Next night, I didn't sleep AT ALL. It took my body a few days to finally adjust. Yikes, those first few days were rough. Since then, I try to be well rested before leaving the country, and that has helped greatly.
2) I stay up the entire day and don't nap on arrival day in Europe. So even if I arrive at 7am without having slept the night before, I stay up all day without a nap and make sure I go to bed at local time and get up the next morning local time. Other than the trip above, this usually works fine. I may have some trouble sleeping the first night or two, but I usually sleep at least a few hours and gradually a few more subsequent nights. Some people nap as soon as they arrive at the hotel in the afternoon. I guess that would work for you - not for me!
3) I just accept that I won't sleep well the first few nights and stop worrying about it. I used to get stressed about not getting to sleep or waking up wide awake at 3am some nights, but I've stopped worrying about it. Whatever it is, it is - I'll survive a few nights with only a few hours sleep.
The last few trips, I've stopped on the east coast for a few nights (not for jet lag but to visit family) before heading on to Europe, and I've had pretty much zero jet lag (other than being very groggy that first arrival day), which seems amazing. Five or six hours time change is a lot easier for me to adjust to than eight or nine, it seems.
In the future, I'd try for better flights than those, without those long layovers - yikes! Having to stay at the airport all day and try to stay awake? Doesn't sound fun. If you've never been to the Sixth Floor Museum in Dallas ("book depository" where JFK was shot from), you might have time to sneak over there during your long layover in Dallas - highly recommend it as something to do! Not sure you'd have enough time in London for something similar.