The short of it is, I've got a 5K run coming up in about 5 weeks, and no training. I need to shore up my worst weaknesses just to complete it, but how?

In more detail, I'm 53 years old, in pretty good cycling shape. I ran the shorter distances in track in HS, and then basically nothing until a half-assed 5K about 4 years ago. I did a little running Sunday and today - the run-walk thing - and it's even worse now. Knees are weak (I didn't expect that), groin tightens up, and balance and coordination aren't that great either. I figure the most urgent is strengthening the knees without injury. What's the best method in the time available. Would stairs be viable? Some weight training? Or would you tough it out and just run?

Run. The big thing is the quads, which aren't used to the different contractions required by running when your foot hits the floor. I think its referred to as eccentric contractions vs concentric contractions, but the more knowledgeable may correct me on that. If I return to running after a lengthy break, as I recently have, it takes me about seven to ten days to get past the muscle soreness in the quads and, to a lesser extent, the adductor muscles.

Just run at a very modest pace, preferring a high cadence, short-striding rhythm to look after your knees. Five weeks is plenty of time to prepare for a 5k if you are aerobically fit from cycling. Five minutes four times a week the first week, ten minutes the second, fifteen the third etc. by week five you'll be able to run 5k.

Walking is a perfectly acceptable way to finish a 5K. Anyhow, 5 weeks is a reasonable amount of time to train. You could try a C25k plan and start wherever your fitness level is at.

https://www.c25k.com/

That's what I meant by the run-walk thing, the couch to 5K plan. My knees are unhappy today. I guess I'll have to see if they line out in a day or two, and evaluate from there. I've got a feeling that if I stress them much more or too quickly I won't have any choice but to walk it.

You can always repeat weeks in the C25K program until you're confident that you're ready to progress to the next week.

It is even more important with running to listen to your body. Running related injuries take ages to heal in my experience.

Amen to that

A guy posted on here a few years back that he had to run twice a year in the military for physical qualifications. He said that his knees were giving him problems and he could not train for the running. He decided to bike for his exercise and to only run on the days that he had to qualify. So now he only runs 2 days a year and he has no problem qualifying.

Maybe you can have similar results.

I could have gotten away with that 20 or 30 years ago, and that's what I did for the 5K 4 years ago but it wasn't much fun. Part of committing to a run is to get yourself running.

I'm happy to correct myself: the pain in the "knees" was just normal muscle soreness in areas I hadn't expected so I'm good to go for continued training. I think Chasm54 was spot on, eccentric contractions (or the shock absorbers as we land, for those of us avoiding medical jargon). We don't really use those in cycling, and a few others I suspect. I still wish there were some feasible way to concentrate specifically on those muscles for a couple of weeks, getting them up to snuff and then running.

You could always try adding some core work including balance exercises. Standing on one leg, one legged squats, eyes closed (if you dare!). When you run you use a lot of those little muscles that get neglected on the bike. If you can try to run on softer surfaces like grass/trails which are noticeably kinder on the joints.

I have a painful knee but have found that I can run pretty far by a) disregarding any interest in speed (10-12 minute miles) and b) paying careful attention to landing on my forefoot, the ball of my foot, rather than any part of my foot behind there.

This. Pay attention to your technique and ramp up slowly as mentioned. Landing on your heels acts as a brake to your forward motion and sends a shockwave up to your knee. Check out the book Chi Running, your library may have it.
https://en.wikipedia.org/wiki/ChiRunning

Thanks for the advice guys, but an broken clavicle is putting a hold on my training plans. when I do get back to it, this slowly softly jogging will probably be the order of the day.

A 5K is a running RACE. The idea is to run and race against others.

It's amazing how in the past few years people get the notion to sign up for a race, including marathons, and think it's okay to walk.

Now there are some events with the "walk" title in the name. Or some races have "fun runs"

OP I'd ignore that comment. /\
I race 5ks a few times a year and walkers are perfectly acceptable in every race. They just start at the back with the strollers.