Here's a short routine for improving balance. Also makes a good test to see how different 1-leg balance is between right and left legs. Everyone is assymmetric to some extent. The suggestion is to alternate the routine with or without shoes.
The hamstrings are important when it comes to supporting knees. Strengthening hamstrings does not require any equipment. You can do useful exercises with just a mat. The exercises in the first video take about 8 minutes to finish, assuming you can do them all without any rest in between. The second video sequence takes about the same amount of time, but includes a slightly different set of exercises. The third video has six exercises (5 min for demonstrations) and includes a few exercises I did as part of knee rehab physical therapy. Although all the videos include double and single-leg bridges, they are done differently. The first are done on the floor, the second uses a chest of drawers (instead of a wall or chair), the third is the hardest. I sometimes do bridges using a stability ball, which adds a little bit of core work.
Using TRX as part of knee rehab is how I got started. I found the first video shortly before starting to work with a personal trainer who had experience using the TRX with seniors. She and I learned together about which exercises were most relevant for strengthening the muscles that help support knees.
The first part of the second video shows how the TRX can be used during a physical therapy session after knee surgery. The last two videos are examples of physical therapy for a young athlete recovering from knee surgery, most commonly ACL reconstruction.
TRX - Knee rehabilitation, 1 min
TRX Training Knee Reconstruction Surgery Physical Therapy
https://youtu.be/GytYWmRczFQ, 15 min, TRX in first 6 min
TRX Training: Knee Rehab Phase 1
https://youtu.be/AFJofGKbjyM, 5 min
TRX Training: Knee Rehab Phase 2
https://youtu.be/0mcnrJcSqJk, 6 min
There are some HIIT workouts that are dominated by high impact exercises that leave out the upper body. Here is an example of a workout that engages the entire body. She is using an 8-lb medicine ball but obviously could start with a much later ball and work up the weight.
With any HIIT workout, it's important to warm up, cool down, and stretch afterwards. This video is designed to allow people to follow along, with a count down and audio markers at the end of each period.
15 Minute Medicine Ball HIIT Workout - Total Body Circuit
For a quick look at a way to work on hamstrings with a stability ball (Swiss ball), check out these 1-minute videos. The third video is a seated hamstring stretch. She mentions how to choose the correct size ball based on height because there are three sizes available.
A Swiss ball is one of the items that I find worthwhile to have at home. Often use it during personal training sessions.
For ideas about progression if you want to make the hamstring exercises more challenging:
The variety of exercises and stretches with TRX is one of the advantages of using it. Another is that almost every TRX exercise helps build core strength. Here are a few ways to use TRX for a HIIT workout in ten minutes or so. The second video includes a warm up sequence using the TRX, as well as a cool down sequence. The third video is a quick look at other exercise possibilities to include in a TRX HIIT routine doing six exercises for 1-minute each.
I learned the first sequence with the help of my personal trainer. We modified a few of the exercises the first few times. Partially so I would not over do it because I had not been doing much cardio endurance work at the time. But mostly to minimize the amount of jumping that I tend to avoid because of being ACL-deficient in one knee. I can jump easily, but see little reason to do it just to increase cardio when there are other approaches that work as well.
The BOSU was the first piece of exercise equipment I bought for home use when doing knee rehab. Starting using it the first PT appointment. This 2017 video is an introduction by Bob and Brad, the "most famous" physical therapists on the Internet, of simple balance exercises using the BOSU. They demonstrate progressions from easy to hard using the BOSU flat side up and flat side down.
BOSU Ball: 10 Best Balance Exercises for Athlete and Post Rehabilitation