Your operation could maintain 2 sets of cash flow: one for patient care, and one for administration. Maybe you could qualify for NGO status and get some government to fund some of the administration side.

This would be somewhat similar to Costco's model, where most (all?) of employee wages come from membership card fees, not product revenue. With this model, you could still let people donate 100% of their funds to the patient if they wanted.

You could even make a slider with an animation that displays the percent which goes to each part of your business - so 10% or 15% of the donation could goto administration.

I love your idea, it's tremendous. Good luck!