So, we're stuck, unless we can find a way to follow the cold pipe from the valve-less valve. Clearly, we need a valve. There are two places of interest: the pump and the dome. The pump is missing a cog, so it doesn't work. If the cog is anywhere, it's in the dome. After we get the pump going, we'll just have to wait and see what happens. Oh, and there's also a locked door involved in all this.

Ask Fooruman what he meant by 'relative equality'.
Investigate the boxclaw, and try to operate it.
Go to the pipe junction to the north of the dome. Turn the wheel so that the pipe leading to the cliff is warm. Tell the Mythslayer to pull the wheel out.