Battery help...Battery lights on, Alternator seems to be good.


My mom has a 2010 Honda Accord with 140k miles. Today she started her car and the battery light came on on the dash, then she noticed a little smoke under the hood, which is from between the power steering pump and the alternator. After she turns off the car, the car wouldn’t start back up, like a dead battery.
I came home to work on the car, the battery was dead, The car was able to start up after I jump the battery, but I didn’t see any smoke nor is the battery light on. I took the battery to AutoZone, they said the battery was charging, but I ended up buying a new battery (battery was 1.5 year old from NAPA). Right now the car starts up and runs fine with new battery.
Moving back about 6 months, the car was low on Power steering fluid, and the power steering pump was covered in grease, which caused the car to make loud noises when turning, I topped it off with Honda fluid and replaced the O-rings in the pump. Today I noticed the grease around the pump again, but the fluid is still at the same level as when I topped it off, so it can’t be a leak. I cleaned the pump and the alternator with brake cleaner today. I never found out what caused the smoke and for the battery light to come on.
What could cause the smoke, could it be serious? Should my mom drive the car now that there is not battery light or smoke? Should I just replace the alternator and the pump? Or is it just simply the battery problem?