DOE for Patient Satisfaction

I think you are getting DOE and the subsequent analysis confused.  If you are taking happenstance data in ordinal form (ie likert satisfaction scales) and then running it against various factor settings to determine the magnitude and direction of relationships of various factors on customer satisfaction, then I believe a better description of what you are doing is it running either ordinal logistic or linear regression.
So I think the better question to ask is can you regress historical customer satisfaction data against various predictors?  sure…try it and see if it gives you anything interesting FOR FURTHER STUDY.  The key point here is that you can use what you find to indicated potential relationships between factors, but you can’t say one thing causes another….correlation, sure….influence or causation, nope….not unless you control the experimental environment..ie run the DOE.

This was not my industry, but I will say it is certainly possible to conduct DOE’s where the final assessment of the results takes weeks, or months to get.   In the electronics industry, the impact to the final device yields may take months to see after the design takes place.
In your case, I am more concerned as to whether you will have the discrimination on the Survey data to see the impacts.  If current patient satisfaction is 90%, and success would be measured as 96%, it is going to take a large numbers of patients to find that difference.
Also, you need to make sure the factors are controllable so that the DOE settings are solid, and make sense when the analysis is done.   There is no reason DOE’s can’t be done, though, if they are well thought out in advance of the data collection.