As Tᴚoɯɐuo mentions, there is correct English grammar, and then there is good English style. You have the grammar part, but "work towards developing" is wordy. You can substitute with alternatives like "seek to develop" or "try to develop" which avoid the stylistically awkward -ing form.
Integrity and discipline are traits I work to develop every day of my life
Aside from the basic English style stuff, the real problem here is that your sentence is wishy-washy. I don't think you want to sound indecisive, as if you're trying to do something but possibly failing at it. When making statements of personal character, be bold:
Integrity and discipline are traits I develop every day, (through ...)
Even this is somewhat juvenile, in my opinion. It's easy to claim to have moral values, but what's more interesting is to describe the ways in which you effectively demonstrate these values:
When I was sixteen I promised my father that I would help him take care of my elderly grandparents, so, as I have done every morning since, I wake up early to cook them breakfast before I head off to school.
With this sentence I don't have to say "I have integrity and discipline". That's obvious from the context. It's a much more powerful yet honest way to deliver the point.