How can I easily keep track of my landing count during pattern work?
The trick my CFI taught me is to use your Course Deviation Indicator or ADF to keep track of landings. After your first landing, bug a course of 010. After your second landing, bug 020. You can reach over and adjust the knob after every landing without having to juggle a pen and a notebook. It's still a manual step, though.
You could also use a product like Foreflight or CloudAhoy or even a service like ADSBExchange to count the number of approaches you made after the fact.