Thanks for the great video. It is not common to find videos explaining how to design amplifiers with MOSFETs, in general you will find polarization analysis. To put things on the right side: when you check the value of VRD = 1/3VDD it is to confirm the voltage drop from the value of RD = 35k to ID 0.1mA, which results in VRD = 3.5V. Is correct? After all, with RD = 37.5k you get VRD = 3.75V which is close to 1/3VDD. And regarding the use of 1/2kn in the ID equation, in this case we have kn = k'n(W/L) and thus there is dependence on the MOSFET geometry. On the other hand, it is possible to use only k = ID(on)/(VGS(on) - VT)^2, where ID(on), VGS(on) and Vt are available in the datasheets.
I think it is clear from the context that K means kilo here. This is not a technical document or some sort of data sheet, just a video where I am explaining the design process. cool down.