Is there an easy way to estimate the number of mining nodes? I was curious about the number of mining nodes on the network, yet I haven't seen it discussed.
I tried two different approaches to approximate the count of mining nodes. I looked up the hashrate of the fastest hashing machine (which appears to be the Antminer S19 Pro at 110Th/seconds) and compared its projected mining profit with the total mining reward.
A profitability calculator estimates that I'd earn 0.2911 Bitcoin per year with the device running 24/7/365. This means that at 6.25 bitcoin per block, I contributed to solving 0.0465 block in 1 year. If the network used only these machines, then I estimate 55,000 blocks / 0.0465 blocks per mining node = 1,182,795 mining nodes. This is the lower bound, the minimum number of mining nodes on the system.
Now, my other assumption is that the network difficulty is equivalent to the average weighted hashrate of the sum of all mining nodes on the network (time frame is probably smoothed here). If the current, March 18 2021, network difficulty is 21.6 Th/seconds, then the same calculator calculates that my device would earn 0.0572 Bitcoin in 1 year. Using the same calculation, I get 55,000/(1/(6.25/.0572)) = 6,009,615 mining nodes.
These two numbers are quite a bit apart. Is there a better way to estimate the mining node count?
While it's possible to estimate the current hashrate from the difficulty, they are two different measures. You can find an estimate for the global hashrate on the same site where you found the difficulty. The current estimate is 162 EH/s (162,000,000 TH/s).
Using a similar approach as proposed by you, we can estimate an upper bound of  162,000,000 TH/s / 110 TH/s = 1.47M mining nodes by comparing the global hashrate to that of the fastest mining equipment.