I have a properly notarized Power of Attorney letter for a single relative in Kentucky.
Does that relative also need a Last Will and Testament, or does my PoA cover whatever may need to be done when they eventually pass away?
There are two relevant laws of Kentucky. KRS Chapter 457 is about power of attorney, and KRS 457.100 specifies how the POA terminates, in particular (1)(a) says that it terminates when the principal dies (therefore on death of the relative, you have no power).
If the person is unmarried, the estate goes to the children or descendants. If there are no children, it goes to the parents; otherwise to siblings.
If there are no living relatives whatsoever, then the estate goes to the state in escheat.
A will is therefore necessary if the person wants something other than intestate default. Also, inheritance via intestacy is procedurally inconvenient compared to a simple will saying who gets what.