My car has 61000 miles on it. 8000 miles ago the end of the dipstick broke off
and is at the bottom of the oil pan, probably covered by sludge. Should I continue to ignore it?
Absolutely not , Checking oil level is a thing all vehicle owners should do . any shop can remove it and order you a new one.
If you do your own oil changes. you could get a strong magnet and run it along the bottom of the oil pan and then to the oil drain whole. you probably can get it out with a few tries.
No experience w/your car or engine, so I can’t say one way or the other. But if that happened to my Corolla or truck, I’d just ignore it, and hope it reaches the magnetic drain plug at some point then I’ll be able to get it out. The oil pickup has a fairly fine screen, so unless that part is defective, unlikely to ever get sucked into oil pump or engine b/c it can’t get past the oil-inlet screen.
It is important however that you secure a new dip stick, so you can check engine oil level. Have you done that? If not you may find the replacement won’t go in all the way b/c a piece of the old one is stuck. If that happens, you shop mechanic should be able to easily solve that problem for you.
You should replace the dipstick so the oil isn’t under-filled or over-filled during an oil change.
As far as checking the oil, if the car can’t inform you when you need to add a quart of oil, it isn’t much of a luxury car.