After I enter a new search engine and its URL, how can I actually add it to Chrome? There is no button to push; actually there are no control elements whatsoever.
How to add a search engine:
How to use a search engine:
The other answer is out-of-date. In current Chrome, right-click the address bar, select "Edit search engines..." and scroll to the bottom of the lower list of search engines. You need to fill in the three boxes for name, keyword and URL. If any of these do not have valid text entered, they will be filled in with a grey background and there is no way to add the search engine. Once they are all white, you can press enter to add the new search engine.
There doesn't seem to be a way to add using the mouse, and the new search engine will just disappear into the big long list of search engines once added, so you need to go and find it to check it worked. You can then hover it and click the "Make default" button that appears to use it as a default search engine.
Try "Example", "example.com", "example.com/%s" to test this if you're not sure what kind of values to fill in.
Funnily enough, now the option to right click on a site's search box and add an SE (with popup and three text boxes) is gone and thats how I came to this page.
Apart from Simon's answer, (I cannot believe they would make things manual/worse for us users), also note that now do-no-evil-google also makes sure that when you add a site as search engine, it also auto-adds the site itself with site name as keyword and apparently there is no way to prevent this unneeded duplication.
So remember top clean up your SE regularly from this nonsense. Chrome has decided they will never fix this as non-power users dont know how to add an SE and this is for a good user experience !
To make things easier, add the path to settings with an easy keyword on an SE.
I have an SE with kywd "cse" (Chrome Search Engines) which leads to "chrome://settings/searchEngines"