Long rambling speech pattern. The te-form ~ていて chains an ongoing activity to the main clause ('while I was playing ball, I broke it'). 使っていたもの ('the thing I had been using') is a relative clause with past progressive, referring to something that broke. わってしまった (割ってしまった) is the past ~てしまう form marking regret at an unintended completion. のです gives a polite explanatory finish.