How to build an analytics system

I need to build a homegrown analytics system that logs user events, which will then be transformed, and used to generate reports or allow for ad-hoc querying.

One of the big issues I'm wrestling with is the impact on developers of the web app and mobile app. We already have them in place. So developers would have to go back through all their code to add code to pass in the appropriate event and its properties. This would be a lot of work. I'm working if there is a way to do this that limits the about of code devs have to add to the existing front end code. I thought about pulling server logs but not sure the server logs would have the right event/properties without having to add a bunch of code to the front end.

Yes, I know I should probably being using something off the shelf, but that isn't an option for me (not my decision!!)