Workout Suggestions?

Cycling/biking is great...I'm a bit biased in that I'm a cycling enthusiast and it's my primary form of cardiovascular exercise...but I love to ride and it's great exercise and can be a great workout also depending on how training bouts are structured.

If you stick with the rowing machine, look up resources to learn good technique (Dark Horse rowing, Concept 2 web site (even if it's not their rower, but is the same general type with a sliding seat & single handle), among others). Poor technique limits your ability to get a good workout in the long term, and can increase injury risk. Sadly, many/most trainers at gyms don't have very good technique themselves . . . even ones who are decent trainers in other respects. It's better to get technique right initially, vs. trying to fix it later.

Whether it will work for you is individual specific. (I'd say the same about biking, personally - both of these exercises can create some knee stress even though low/no impact, so it matters what the pre-existing problem is. If you can bike without problems, that increases the probability that rowing will be OK - similar straight-line hinging motion - but no guarantees because there are also some differences.) I have a torn meniscus and some knee arthritis. Things with twisting of the knees or much impact can be a problem, but both biking and rowing are OK, as long as I manage workout schedules sensibly to allow recovery, and phase in volume increases gradually. As background, I've been rowing on water and machine for nearly 20 years, biking some for longer, took spin classes for well over a decade before the pandemic.

yay biking! Hopefully your gym has spinning bikes available for non-class use, cos the normal exercise bikes are horrific. If they have one try the recumbent bike it's much comfier than the usual sit-up-and-begs. And try spinning but take it easy til you see how your knees manage it.

Can you skate? It's low impact, and you can do the same sort of drills that you do for running. If you've *kitten* ankles I recommend quads over blades. If money isn't an issue check out rollerderby quads, but honestly anything will do. If you get cheapos you might need to fork out for better wheels but they're worth their weight in gold, and you can buy skates and wheels second hand, also get decent padding (TSG are my fave but are pricey).

I also agree with sijomial on the weight training for knees, it helps mine out alot (knee issues from back muscle damage and pelvic tilt issues). Also i'd say Sports-Chiro over physio but that's from personal experience.

Also what about yoga, pilates, and traditional martial arts?
Tai chi or maybe something quicker like shotokan (anything that doesn't focus on kicking, so no tae kwon do or muay thai). If you can stand in a sumo squat, yoga warrior stance, and wall sit for a minute without it affecting your knees you'd probably be ok at karate. The sensei will take your health issues into account so no need to worry about that - if they don't find a better dojo.

If you decide you like rowing, maybe check to see if there's a local rowing club? If they're nice they'll have beginners courses and do taster sessions, if you join you'll get technique help.

Also what about HEMA or olympic/sports fencing (foil, epee, sabre)? HEMA doesn't need to be high impact, it's very like doing karate tbh, I do longsword but you have loads of other disciplines that might be more your style. Olympic fencing might be beyond your knees but you could always try wheelchair fencing if the local club offers it and has a club chair to ensure actual users have more than 1 partner to fight (some do). Most fencing clubs do taster sessions, I thoroughly recommend stabbing or bludgeoning people with swords... I mean learning new skills and maiming people.. no that's not it...
Both can be expensive but it's the kit that costs, and most clubs will have club kit you can loan til you get your own gear (buy your own helmet first if you join up cos eeeeeewww).

Good luck with whatever you try.

What about seated/chair HIIT workouts? I do not know anything about these types of exercises, but it might be worth looking into them as a possible knee friendly cardio workout.