Difficulty finding research mentor
This suggestion is totally "off the wall" and "outside the box", but the underlying idea is that you broaden your perspective. I'm assuming a few things. First, you are at a top institution. Next, you are a top student - you are obviously dedicated. Next, you aren't alone and there are at least a few of you in the same circumstance. 

Consider forming your own "research group". Become its leader. Find a faculty member who will give you a problem to work on and maybe give you occasional ideas on your progress. The commitment of the faculty member can be minimal and your research discussions are mostly among group members. Find someone, maybe even a postdoc, who can give you some ideas about research process in your field. 

But if four of you approach a professor it is a different sort of thing than going alone. A junior professor might be an especially good "target". 

Try to have your group meet a couple of times a week to share ideas and what insights anyone has. 

Produce a group report at the end of some period of time, even if it only discusses directions, not successes. You can even discuss failures, since it was learning that was important and it is just as good to know what doesn't work as what does in many cases. Offer the report to some faculty member and ask for comment. 

This will have two benefits. One is that it should enable you to learn something about research process even if it comes with small progress. Knowing how to go about research is an important skill in many fields and it is good to get some practice at it. But having formed such a group is, itself, a big plus when it comes to talking about grad school application.