How many of those things are used in the video ? I think 8.
The “BlazePod” seems to use 6 of them and costs hundreds of dollars.
For the light, you can use a single LED or a RGB led for 5V, or a 12V led with a mosfet to turn it on.
To read the touch, perhaps a FSR will do (Force Sensitive Resistor 0.5" - SEN-09375 - SparkFun Electronics), but a switch (or a few) is probably better.
Can you make a mechaninal drawing ?
It could stand on three feet. There has to be a switch (or three switches) that can be pressed.
The sketch is not hard. Use random, show the led, read the input, even calculating the average reaction time is not hard.
What troubles me most is the exercise itself. That can’t be good. In my opinion such exercise is bad for your body. It is bad for knees, ankles, spine and what else. Can you show it to a physiotherapist ? I think the constant bending towards the ground with force is not good.