[Radioroslyn]

Here's a thread of restoration of a 16X that includes the reconstruction of the speaker cone. The speaker is a bit huskier than your K type, this one is a 10 1/2 U speaker. https://philcoradio.com/phorum/showthread.php?tid=17572\

One of the most time consuming parts of reconing the 8'' is finding the proper 8" cone. There are hundreds of replaces but one of more of it's dimensions will be incorrect. In short you'll do a lot of looking to try to find something that may work.  You probably want a paper cone rather that a paper with a foam edge. The foam edge budges out and you'll need to add a spacer between the cabinet so the it can move freely. 

I've reconed a handful of early K from 1930/31. These are different from your spkr, the cone is deeper on the early ones. I use the old vc/spider and glue it to the new cone.

If you are going to diy cleaning is important. The inside of the frame and the gap (the part that the vc fits down into). Dust and dirt gets down in there and impedes the movement of the vc which in turn limits the frequency response.

GL