Extraction and GC-MS or maybe LC-MS, if you really need to measure these specific impurities
I don't know a lot about this area, but you are looking for very low amounts of impurities. For ethylene oxide, for example, your desired detection limit of 0.07 mg in a 10 g sample corresponds to 7 parts per million, or 7 ppm. Your limits for the glycols correspond to 100 and 400 ppm.
I'm not an NMR expert, but I think NMR will not be remotely sensitive enough for this purpose. Even if pure, NMR on a 0.07 mg sample is challenging and far from routine.
The best route forward might be a selective extraction of these impurities. You should chose a solvent that does not dissolve polymeric PEG at all, or as little as possible. This can be a challenge, as PEG is soluble in water, alcohols, and even dichloromethane. Ethylene oxide is a relatively nonpolar gas, so you may be able to dissolve it in hexanes or a similar nonpolar solvent. Unfortunately the glycols are unlikely to be soluble in hexanes.
So for ethylene oxide, say you extract 10 g of sample with X mL of hexanes, and say X is ~100 mL. The resulting solution would be 700 ug/L of ethylene oxide at your desired detection limit. That should be detectable by GC-MS.
For the glycols, supposing it is somehow possible to find a solvent that selectively dissolves the impurities but not the polymer, you could then derivatize the solution of glycols with a reagent like TBDMS and analyze them by GC-MS. Alternatively, you may be able to analyze them without derivatization by LC-MS, but sensitivity of electrospray-based LC-MS for aliphatic alcohols is limited. GC-MS is probably the way to go...if you can find a solvent for the selective extraction.
If no selective solvent is available, you might be able to analyze the glycols by directly derivatizing the solid PEG sample with a reagent like TBDMS in a liquid solution of some (non-selective) solvent. The derivatization will convert the glycols into trialkylsilyl ether derivatives. These may be lower polarity than PEG or more volatile than PEG such that they could be selectively removed and analyzed after derivatization.
Do you really need these specific impurities?
Maybe thermogravimetric analysis (TGA) would work for you. There are variants of TGA where a mass spectrometer samples the gas phase above a material while it is slowly heated and decomposed. If you have a sample of PEG you know is "pure enough", you could use that as a reference material, and then spike it with known amounts of contaminants to verify that these contaminants result in a perturbation to the TGA signal, or to the TGA-MS signal.
You may or may not get a precise numerical estimate of the mass fraction of your target impurities, but you will probably get a sense of whether a given 10 g sample of PEG is pure "enough" as measured by similarity to known reference material.
Analytical chemistry is hard
No matter what, you'll need to do a lot of methods development work to verify the appropriateness and robustness of your method. E.g. what's the best value of X to use for selective extraction? Can you for sure detect the impurities at your target levels with your chosen procedure? You'll probably need to spike reference material with known amounts of impurities to make sure. And is the response of your analysis method or instrument linear with respect to impurity mass fraction? If so, over what range?
These are things that analytical chemists spend a great deal of time fretting about. It's not easy.