Why does "how" not follow the structure of the other interrogatives?

Solution 1:

I can confirm that there does not seem to be any corresponding 'th-' word for 'how' ('how is a 'wh-' question word).

As far as I can tell, the corresponding cognates in German, French and Latin (with cursory investigation of other IE languages) for 'how' are also lacking a demonstrative counterpart.

As to reason why there is none, as alluded to by the others, one can only speculate.

"How are you going to get to San Diego in time?"

"I will get to San Diego...by taking the freeway."

The answering phrase is a prepositional phrase, whose question would have been 'By what way...'.

Given that very particular example, I'd guess it is because, given that 'how' covers too many possibilities, too many 'manners' could be answers, so there's no simple one word response indicator.

Solution 2:

I think user9325 is asking about the "when/then" patterning, rather than the "wh".

It's rarely possible to give a satisfactory answer to "Why doesn't X language have Y feature?" Languages are not designed by committees, or engineered: they just happen. Some patterns exist, but they may have exceptions or holes.

Notice that beside the examples you give, we have "whither/hither/thither" and "whence/hence/thence", but these are obsolete or obsolescent: one might just as well ask why they should be falling out of use.

I'd never linked up "what" and "that" in that way; and I'm not sure that isn't a coincidence ("what" goes back to IE forms in '-d', but I don't think "that" does); but in any case there is no "hat"; and "who" does not enter into this pattern at all.

Not a real answer, but as I say, I don't think there is one to be found.

Solution 3:

Well it's not obvious to me that how is particularly different.

If you're just wondering why it doesn't start with a w, you might as well wonder why we don't pronounce the w in who, for example.

This from Merriam-Webster...

Middle English, from Old English hū; akin to Old High German hwuo how, Old English hwā who — more at who.