Analyzing percentage data that clusters around 100%

I have data from 10 mice looking at the percentage of a time period that each subject spent immobile before and after various drugs. I want to be able to identify any effect of drug on immobility. The data is in percentages because the time periods observed before and after drug injections are different. Most of what I've read about analyzing percentage data recommends using an arcsin transformation or something similar. However, most of my data points fall in the range of 90-100% with a number of 100% values, making a transformation less helpful. I also have the added complication of repeated measures (each mouse received each of several drugs). I would greatly appreciate any ideas or suggestions for how to deal with this data. Thank you!