I would definitely suggest adding like a few more equivalent of TsCl. I've played this exact game before, tosylation of polymers is just a crummy reaction. I'm sure every organic chemist at your school has a big old' 500 g container of it sitting around to get from. Neat pyridine isn't a bad idea, but if you feel like nothing is dissolving, adding some DMF and heating it a bit (remember pyridine's BP!) should help.
That workup would work for a small molecule, but is doomed to fail on a polymer. I bet it came from a small molecule guy. I'll give you a crash course in polymer workup from someone who has made a couple hundred novel ones.
Polymers virtually always need to be precipitated and redissolved a few times to clean up. Small molecules become caught in the polymer matrix, so just doing liquid liquid extraction won't fully transfer contaminants from one phase to another like in a small molecule reaction. Also, polymers make great surfactants, so extractions can easily emulsify.
Once you think the reaction is pretty complete, take a drop of it and put it in some cold ether, methanol, and isopropanol. Pick the one the precipitates best. I can't explain exactly why, but those 3, (plus water, when appropriate) are just really terrible solvents for polymers, but decent ones for small molecules. Hexane could work in this case too. You should see a solid precipitate out that is not a very fine powder. This judgment unfortunately requires some experience with polymers, but you want the precipitated solid to be easily captured on a filter and able to be redissolved and precipitated again. 3x is typical. Each cycle should drastically reduce the amount residual pyridine/DMF/TsCl left. It may seem a little silly, but I would smell your solid. If you can detect pyridine, keep precipitating. If you smell pyridine, the other two are likely still there too. Often each cycle will lower your yield, but it's worth it for purity.