Currently living in a place with c. 35º temps and 85%+ humidity.
But I'm not going to tell you to suck it up, because I have a/c and am miserable without it. When our power cut out during a storm, I lasted about four hours before I risked a quick dump of water over my head. (Couldn't take a shower during the lightning storm.)
However, this has made me a bit of an expert on miserable thick heat.
First, I second the idea of a cheap hotel with a/c, just for a couple nights.
Barring that, if you can't/don't want to get a cheap window a/c, I would recommend a dehumidifier. They're also useful in the winter when it gets damp, so you won't feel like you have a temporary appliance. In fact, it speeds up laundry drying if you don't have a tumble drier--a key UK appliance, if you ask me.
A dehumidifier will suck the damp right out of the air, and then the ice-in-front-of-a-fan will feel so much better. You can get a seriously powerful one for under £150, or if that's a stretch, a less powerful one for £50--and it will deliver the next day from Amazon. So, for example, Humidifier . Or, if you'd rather something smaller, Smaller .
You also need a decent fan, not the crappy rotating ones. I use a Vornado, which is incredibly powerful. That with some ice in front of it will feel amazing, particularly if you mist yourself first with a spray bottle.
Honestly, once you suck out the moisture, it'll feel so much better. It will still be achingly hot, but it won't feel so thick.