I'm wondering what the proper way to clean the tub is with chlorhexadine?

http://www.reptilebasics.com/chlorhexadine-16oz

I diluted the solution to about 2 tablespoons to half a gallon of water.

I just soaked a paper towel in it and wiped down the inside of the tub and items in the tub with it... and then soaked another paper towel with water, and wiped it down again. I'm just making it up as I go though. Is the solution harmful if it's not thoroughly washed out in the tub? Can I clean the snake itself with the diluted solution?