Ok, first thing is don't fret. 13 weeks is still early for some puppies. I know some are toilet trained by then, but there are plenty that aren't so don't think you are an exception. In addition, you have the added difficulty that he has had an upset tummy which has made it even more difficult.

Keep on taking him out after every meal, when he wakes up, after play etc and he will get it eventually. It may be worth removing the hall mat for a while, and make sure that any messes are cleared up with biological liquid to remove any scent.