What cadence should you sprint at?

I'll add a few things in addition to @Criggie's answer:

1. Power meter. It was mentioned in the comments, but you really don't know if you produce more power pushing a bigger gear with more force at lower cadence than you produce with a smaller gear at a higher cadence unless you can measure it.

2. Expand your usable torque range. Everyone knows about things like anaerobic intervals of 3-7 minutes each to train your ability to ride anaerobically for several minutes or 10-to-30-second all-out sprint intervals. You can do the same for cadence, too. For example, 3x7x3 intervals at 100-110 rpm where you do 3 sets of 7 minutes of riding at 100-110 rpm with three minutes of rest between intervals. Or a 20-minute interval at low-to-moderate power where every 30 seconds you spin out, maxing your cadence to the fastest you can do seated. If you've never worked on expanding your effective cadence range, you'll be surprised at how painful those types of rides can be even though your power level or heart rate never really gets too high. If you're doing power-based training, the power level on these rides will be pretty low. But you'll hurt anyway. The high cadence can cause your HR to elevate over what you'd otherwise expect for the effort level, though.

To expand your ability to generate power at lower cadences you need to build muscle strength. For me, the most effective means of doing that were 5-10 minute intervals at 50ish RPM and high z3/low z4 power levels.

Leg strength for shorter time periods for me was best improved in the weight room.

Really short-term (under 5 sec) and standing-start power at low and even medium cadence is all about pure strength. It's also almost useless in cycling except for track cycling standing starts. Heavy leg presses, leg extensions, leg curls, and squats if you want to go that far worked for me here. This is where how much weight you can max out at matters. But again - it's almost irrelevant for cycling. (And you'll add muscle mass without adding to your ability to generate power aerobically, which is detrimental to your climbing...)

For medium-length sprint effort low-cadence strength, I found that rapid sets of squats with a light weight worked well. Pick a weight you can squat about 20 times in a single set. Now, do sets of 10 reps. With only ten seconds between sets. Do 4-6 sets, or until you fall over. That workout will kick your butt and seriously improve your sprints in the 20-40 second range at any cadence, but especially at lower cadences. Because you're using a light weight there's no real long-term negative effect on your back. Just don't ever do that workout except in a real squat rack with safety bars to catch the weight or with at least two spotters that you trust.