Similarly to enlarge, enhappy means enlarging happiness; becoming or letting someone be more happy than he or she is now.
There is a special web-site created for this word: http://enhappy.com (this is not an advertisement, but the source).
The difference between "enhappy" and "make happy", is that there is no way to "make" someone happy. You may just add some good news, which will enhappy someone.
- I enhappied him by inviting him to the party! (he's grateful to me for being invited)
- Will I make him happy, by giving him as ass-kick? You will make him fly probably.