Single word: extremely frustrated, but relieved

You might describe that feeling as bittersweet:

being at once bitter and sweet; especially : pleasant but including or marked by elements of suffering or regret

I'd suggest catharsis/cathartic or excoriation/excoriated. Both imply liberation from negative emotions, coming at the cost of an emotional/physical outburst.

Answer: No

You've certainly identified the perfect use-case for such, but I don't believe there's any single word which truly captures the described mix of exhasperation and elation.

I say it's the perfect use-case because both feelings are directly caused by the same single source (finding the solution) and are inseparable despite being near polar opposites. Perhaps the reason no such word yet exists, is that before the information technology era the closest possible analog to your use-case is spending too long looking for something, upon the discovery of which you realize you should have immediately known its location. While that should be a pretty close match to the feeling by type, it falls short at least a couple orders of magnitude in intensity, not to mention being a far less common occurrence.

Perhaps now's a good time to fill in that linguistic blank.