As I said above, true intelligence is not possible without consciousness, though these are different notions indeed (as thought and mind). If we assume the existence of intelligence without consciousness, we inevitably expose ourselves to the issue of purpose. That is, what is the purpose of this intelligence? And the purpose of intelligence cannot stem from intelligence per se. In this way, purposeless intelligence is an oxymoron, and it is mind that provides purpose to intelligence. In other words, intelligence is a device of mind for reaching its ends. That, simply put, sums it up.
You are right that in order for even a problem to be declared, and a solution to be declared, a purpose needs to be defined, and purpose means consciousness (because "good" versus "bad" experiences). However, consciousness is only necessary to DEFINE the problem, not to solve it.
As such, you need a sentient being to RECOGNIZE intelligence.
I call something intelligent if it can SOLVE a problem (as DEFINED by a consciousness).
That is: an purpose is necessary to define a problem and its solution:
"the addition of two numbers". In order to define that, you need to say that there's a purpose in the notion of "addition".
However, a thing that can PERFORM the addition is intelligent in my view. A hand calculator has a certain amount of intelligence (but probably no form of consciousness, although we never know it).
Once, as a conscious being, you have recognized a problem with a purpose, you can recognize any system that can solve it, and as such, declare it to be intelligent.
Once, as a sentient being, you've recognized a system that is intelligent, you can just as well ASSIGN IT A HYPOTHETICAL conscience, for which its good feelings are "solving the problem" and its bad feelings are "not solving the problem". Because you can never know, so you can arbitrarily assign subjective experience to just any physical system.
This is why you can, if you want to, assign subjective experience to a calculator, who has "good experiences" whenever a calculation is performed correctly, and 'suffers' when it is not. Whether these experiences are really subjectively lived or not, is impossible to know. Most people would think that a hand calculator doesn't really "experience feelings", but there's no way to know.