The phrase "a sort of joy" is correct and usable in written English.
It can be used to describe a feeling of happiness that is not fully defined or is somewhat ambiguous.
Example: "After receiving the good news, I felt a sort of joy that I couldn't quite put into words."
Alternatives: "a kind of happiness" or "a type of delight."
"There is a sort of joy when the whole audience stands up for something," he admits.
All of us would like to do this, and such feats leaven the film's violence with a sort of joy.
"At times I would wonder why his face would light up with a sort of joy when he'd play something that was so sad it brought tears to my eyes," she observed.
… It might just be my constitution, but I have a sort of joy, enthusiasm and kind of endless reserves of energy for work.
When Ogwumike hit a jumper in the paint to end the run, the home crowd erupted more in relief than with any sort of joy.
You can tell which is which by his expression -- a sort of smiling joy, or a crude crying jag.
Her voice is singular – besides its octave-spanning qualities, it's filled with a sort of holy joy.
Allen's own special brand of nihilism, as expounded in his films, is, of course, well known – equal parts despair and a sort of despairing joy – but entering his office makes me wonder if this is more than an intellectual choice; if he simply has a different physiology from the rest of us.
Experimenting with this always gave me a sort of childish joy: once you start learning the actual optics and wiring of the visual system (as I did when I was studying it), you lose all compunction about playing with it and viewing your own mind's mechanics in action.
They find a zest in this difficult clinging to truth, or a lonely sort of joy in pressing on the thorn..
But for more casual participants, there's a certain sort of joy in watching your luck come together as a Madder Bomber deals all six of his damage on the one big enemy threatening your game.
