I don't have an HQ but most LA are set up similar. You mention you broke the needle did you find the broken part? the tip isn't sitting down in the race area somewhere is it? I would take the bobbin out of the bobbin case inspect everything carefully. Hopefully the machine has a flywheel somewhere that you can manually lower the needle a little at a time into the race area to ensure the race is moving properly and nothing is stuck. Do this without the case inserted so you can watch to ensure everything is moving properly. If you still can't get the bobbin case back into the race area it is possible you through the machine out of timing when your needle hit the improperly inserted case and broke. However, I have never heard of the timing preventing you from snapping in the bobbin case.
PS the case will ONLY snap in one way so sometimes you need to rotate it slightly clockwise or counter clockwise to line it up to properly snap in. It is a very definite click when it seats properly. You hear it and you feel it.