A new engine cost as much as a completely new mower these days.
I suggest Ebay for a used one but ask how many hours are on it and if there is any warranty. There is lots of junk out there on Ebay. The 13HP Kohler was used mostly on JD riders as I recall.
Another source might be a small engine shop if you have one in your area. They are getting harder to find these days because no one wants to pay to have their equipment fixed or maintained anymore.