I was wondering what I could do about the problem I'm having with my big right toe.
What's seems to be the problem?
If there is such a contraction, what is it? What has seems or what is seems? Or is it a typo?
What's is the contraction form of What is. So to expand your phrase:
What is seems to be the problem?
does not make sense and is not grammatical. You could say:
What seems to be the problem?
What does the problem seem to be?
If it was a spoken phrase, maybe it was misunderstood? Where the " 's " was actually the "s" in seems?