The header on the Sense HAT is really cool - it is great for stacking boards miles high.
If you want to solder them, providing they fit on the pads on the board, you should probably use some solder paste and a heat gun (being careful not to melt the solder already on the board). It's gonna be hard work with a soldering iron (I would melt solder onto all the pins on the board and try and reflow it pin-by-pin).
The Sense HAT's PCB was possibly custom-designed around this header, so it might not even be possible. Not all SMT female headers are push-through - most are just like normal but with SMT pads instead of thru-hole pins, I believe.