How do we know that compressing the knee generates a feedback signal that makes the leg extend even before the signal reach the brain?
Q: How do we know brain Is not involved in knee reflex?
A: The brain is involved in the knee reflex in the control of how fast/slow and strong/weak the reflex occurs, but it's not essential in the occurrence of the knee reflex. The knee reflex is still present even when the spinal cord is cut or destroyed at any level above L2 , which is above the reflex arc pathway (L2-4) and which cuts off the connection between the brain and the reflex arc. This is usually evident clinically. The only exception that the knee reflex may be absent after the incident is that the condition called “spinal shock” occurs . However, this condition is temporary, and the knee reflex will return, usually even brisker than before the incident (due to the absence of inhibitory control from the brain). This confirms that the brain is not essential in the knee reflex and only has regulatory roles.
Q: How do we know that compressing the knee generates a feedback signal that makes the leg extend even before the signal reach the brain?
A: Electrophysiologic studies show that the latency of the knee reflex is normally about 46 msec  while the evoked potential (P37) recorded at the scalp over the leg area after electrical stimulation of the posterior tibial nerve is about 37 msec. So, comparing data from these comparable but not identical experiments, it can be estimated that the knee reflex occurs, indeed, after the signal reaches the brain, as you asked. But don’t forget that the former latency is the time of the complete loop while the latter latency is of only halfway. It will take longer than 37 msec for the signal to complete the loop of leg-brain-leg and move the knee.
Xu D, Guo X, Yang CY, Zhang LQ. Assessment of Hyperactive Reflexes in Patients with Spinal Cord Injury. BioMed Research International. Vol 2015, Article ID 149875. http://dx.doi.org/10.1155/2015/149875
Wikipedia. Patella reflex.
Atkinson PP, Atkinson JLD. Spinal Shock. Mayo Clinic Proceedings. 1996 Apr; 71(4): 384–389.