I've just finished rebuilding our "3 season" room, and did a bunch of remedial waterproofing. While I can't give you precise advice on this location, I have a material recommendation for you:
Zip system's Stretch Tape
They recommend using it along window and door framing, and also utility penetrations. It is quite flexible, stretches like crazy and is tenacious stuff. It is also very pricey and smells like cancer. That said, it is very effective at what it does: waterproofing.
If you have an area identified where a flexible membrane would be a good solution, this stuff will deliver.
Thank you.
Also. Those leaf proof gutters aren’t helping, they puddle a bit of water on top. Clean them off. You could also get yourself a piece of galvanized metal (thin), a decent hand cutter (I like Milwaukee) and form a 3 side piece to patch over that, being a top left and right side. Slide it under the roof, use lots of high quality 4th gen silicone that works on wet surface. (Buy the priciest tube). Cover the whole inside surface of the patch, slip it in, bit more goo on the edges. Probably never leak again. May need to open the roof internally and get a dehumidifier goin to get the water out before mold sets in or plan on replacing that entire section of room/roof next year. Prompt attention to leaks yields best results.