Many thanks for the advice and input from our readers about my personal ongoing headache caused by a negligent landlord. As many recommended, I sent a formal and certified notification of the repairs needed via postal service since we have yet to be contacted back. By chance, another tenant had contacted the LAHD today; their downstairs apartment suffered serious structural leaks that became apparent during this past wet weekend, so I was able to talk to the inspector again just now about our broken window...
At this point, we can't do much except wait while the case moves forward. The housing inspector said our landlord has several cases open against him about negligence, repair issues, and is even ignoring the calls from the LAHD ("I went to several local problem buildings and noticed they all belonged to the same owner..."). He agreed we've followed necessary procedure, with documented notice and photos of our broken window (he took more photos), but that we should wait about doing anything further like hiring our own repairs or withholding rent since he's moving ahead with requesting a court case against our landlord for violations, putting a bit more serious pressure on him to fix these mounting problems.
One of our neighbors just came by and warned us about their own rent withholding/deduction experience. They were evicted after pursuing this route, mentioning that state laws seem to side and protect rental owners more diligently than one would expect, even in light of negligence and structural hazards. Their case was a bit more debatable, as it happened to involve interior mold growth, and their mold test results were contested by the landlord's own hired testing in court.
But this definitely gives me more reason to move very carefully in trying to alleviate my apartment's issues without endangering my living situation. We love our hood, neighbors, rent price and the apartment itself, despite the issues. So now we just wait...and sadly, maybe start looking around for another place to live.