Deciding whether the head of a TM remains in K slots?
Hint: Given a Turing machine $M$, you can construct another Turing machine $M'$ with the following properties:


If $M$ halts then $M'$ halts.
If $M$ doesn't halt then the head of $M'$ eventually reaches the $K$th square from the right for every $K$.


This construction can be used to show that the halting problem reduces to your problem.

As you mention, given $K$, your property is decidable. This shows that your problem is $\Sigma_1$-complete.