Why not just use one of the many vinyl washing solutions that have been specifically developed to clean and preserve your vinyl rather than taking a chance? The problem with dishwashing soaps is that they are , well, soap. Usually that translates to a residue being left behind, and residue can end up gumming up your stylus.
I use and can recommend the Disk Doctor products, but there are a plethora of others manufactured by folks who have developed it specifically for this task. It's not very expensive.
Richard, your process is very similar to what I do, except that I then follow up with another cleaning cycle and a good vacuum on my VPI HR 19.5 cleaning machine. The VPI isn't very expensive and it's a good machine to have if you are going to be a serious collector Fish.