Given the requirements as guessed from the description, your reaction will produce the degradation products. The better option would be producing a tethered COOH group to the polymer, although methods for direct conversions of PEG's -OH are available. The ID could be done through, simply, by IR/FT-IR. Chemical and physical purifications are the ways to deal it. The derivatized products can easily be purified (for very small MW PEGs) RP-HPLC, others might need gel filtration with selection of appropriate gel material, usually Sephadex with required porosity providing beads which depends on the MW of the polymer. 

Did you initially heated the solution? Usually KMnO4 need the presence of Mn2+ ions in solution to catalyze the oxydation of organic compounds. Perhaps you can try using a little of H2SO4 to have acid solution and heat untill 60°C and then stop heating and let react overnight. 

How long is your PEG? Are you sure your COOH groups would not fall into the noise level? You may try to compare to a more robust method, for example Jones reagent, to have a baseline for comparison. I expect KMnO4 is generally somewhat too reactive to perform the transformation you are attempting without side reactions. Consider something like TPAP; that should be a lot easier to get working.

Thank you so much for your answers. MW of PEG is 3350. I choose KMnO4 as an oxidizing agent because after reaction I can separate MnO2 from the others by filtering and I think it makes my purification easily. I have so much PEG-OH and KMnO4, so I want to use them firstly. If I fail, then, I will buy different reagent for oxidation. I also saw on some papers that they reached 60 C, but I even cannot reach 25 C. So, I did not try heating, on the contrary I worked on ice because the solution is overflowing. I do not know that I have the degradation products. So, I think I should apply TLC to see purity. If my started material degrade, changing reaction time can be enough to solve this problem for example 3h or 6h instead of overnight. What do you think? I want to ask one more question, Can I use column chromatography by using silica gel for purification of COOH-PEG? I think COOH-PEG will be slow from PEG-OH. Am I right? Because I only have sephadex G100 and I think this is very big to PEG.

Thank you

You could try various dilutions, reaction times and temperatures, and see if anything works better. Remember you can quench the reaction with something like sodium bisulfite solution! If you see your reaction overflowing, I think that is probably a sign of overreaction (you would be forming CO2 right?) As you are doing the reaction in water, I don't really think column chromatography is practical. You may try multiple extractions with dichloromethane to get the polymer in the solvent phase, but getting dry product will be a nightmare.

If you cannot rise temperature, than usr Mg2+ salts as catalyst, Rising temperature is done just to have formation of Mg2+ ions in solution.

For separation you can use extraction in mild basic (ex. NaHCO3) water/Ethyl or Buthyl Acetate. In this way you can extract excess PEG-OH in organic solvent and PEG-COONa in water, then acidificating with HCl you can extract PEG-COOH in organic solvent and simplify concentration of PEG-COOH.

OK, I understand.

thank you I will try it. 

I will suggest you to use succinic anhydride to add COOH group to terminal OH..