How do you decide what an acceptable p-value is for a K-S test? Let's say you're thinking about running a hypothesis test (like a t-test) on some data that requires your sample to be normally distributed. You decide on a 95% CI and you run a Kolmogorov-Smirnov test to find out if the sample fits a normal distribution.
Unlike with your hypothesis test, which you've already decided needs to have a p-value of less than .05, with a K-S test you're looking for a high p-value that indicates a high likelihood that the difference between the distributions is due to random chance. 
It seems intuitive that in this case you would only accept p-values for your K-S test that are over .95, since it would establish the absence of a difference between distributions using a standard that is congruent with the one you chose for establishing statistical significance. 
Is this a best practice, or are there situations where you would accept greater or lesser uncertainty with your K-S test than with your hypothesis test?
How do you decide what an acceptable p-value is for a K-S test?
 A: Size of p-value from KS test is not a proper measure to check the validity of t-test. 
Imagine that the true underlying distribution is not a normal distribution but very close to a normal distribution (e.g., having a very tiny bump on right tail). If your sample size is small, you cannot detect this small deviation so your p-value is probably large. However, if you collect more and more data, your data have more power to detect even such a small deviation, and thus p-value is getting smaller and smaller. Therefore, if your sample size is large, even if p-value seems to be small (e.g., ~0.1), the underlying distribution can be very close to a normal distribution and thus t-test can yield a statistically valid result. (In fact, t-test is robust to the violation of normality assumption once the true underlying distribution is symmetric, see Efron, Bradley. "Student's t-test under symmetry conditions." or Martins, Joäo Paulo. "Student t-statistic distribution for non-Gaussian populations.")
Therefore, instead of using KS test, you should check the histogram of your data and Q-Q plot. Also, you can also use distribution-robust testing procedures (e.g., permutation test, concentration inequality based test) together to see whether these distribution-robust tests give you the same answer as t-test. 
