I have it on a slow charger now, showing 4A.

 

I'm assuming I'll need to buy a new battery. Anything I should know about installing?

 

In my humble opinion the battery is kaput!

If the battery is really 11 years old, you did very well. As most batteries usually last between 4 and 5 years in most cases. I would just replace the battery with a similar capacity battery in cold cranking AMPS and class size of the battery.

A local auto parts store should be able to help in sizing the correct battery for your car.

 

Thanks for that suggestion. I will look into that option. I see there's an odd hold-down bracket with a little module on top. Not sure what's that's for. See top center.

All my batteries seem to last around 10 years. I had a Volvo battery that went for 20.

 

That's some kind of ambient temp censor, similar to the one in the front bumper. Just remember to plug it back after replacing the battery, otherwise the orange service light will lit.

 

Read through here:

Found it! 12v battery!

My 2012 Sonata Hybrid is going on 6 years old (purchased Jan 2012) Can you believe im still using the original 12v battery! The heat down here in South FLorida is hard on batteries and I always get a new one after 4 years. Now that my OEM battery is nearing 6 years old I knew I had to replace it...

www.hyundai-forums.com

 

Tried to start my sonata today and nothing happened, just a few fluttering red lights on the dash. This is after a 250 mile road trip with no issues, so should have been fully charged.

Tried to hook up a 12V charger but the needle went off the scale and the cable started to melt, so pulled it off. I assume that indicates a short in the battery or electrical system.

I checked the voltage with a good meter and it showed very low, about 6V.

Of course the car won't start, but I've got it parked in such a way that a tow truck would have a lot of trouble moving it.

Any suggestions would be greatly appreciated! I'd rather do what I can before handing it over to the stealership.

Is it a simple job to install a new battery? The location is pretty tight.

Replace the battery

 

Tried to hook up a 12V charger but the needle went off the scale and the cable started to melt

I have it on a slow charger now, showing 4A.

So you disconnected it from the car in the second case.
Same charger for both cases?
If so the short is in the car.
What voltage did the battery charge up to?
Be careful putting it back in the car.
Connecting a charged up battery to shorted car electricals could get ugly.

 

Update:
I bought the HR6 Interstate battery today and got it in with no trouble in about 10 minutes. Cost me $241 + tax in Mississauga. Car now starts fine.

Many thanks to everyone for all your advice and comments!

One thing: the Door Ajar light is still on. Wondering if that might have drained the battery? It was odd how the battery just went completely dead, with no warning at all, a day after the door light started staying on.

 

That could be part of the issue you have. Check you door switches. Sometimes they can get stuck.

 

If I press the rubber switch button in the door jamb, the Door Ajar light goes out.

Closing the door normally leaves the light on.

 

Sounds like you found the issue to the dead battery. Is the switch loose perhaps

 

I just went out to check the Door Ajar light and now see that the red light at the bottom corner of the door won't turn off when the door closes. So I removed the little bulb.

The door switch was not loose, just needed to be pushed in further to turn off the Door Ajar light. Nothing I could do to turn off the red light at the bottom-corner oh the door.

Meanwhile I've put the original battery on a charger to see if it really was defective.

 

After charging the "defective" battery for 24 hours I did a test and it's well into the Green on my load meter.

I guess I didn't need a new one after all...

Surprising how the Door Ajar lights can rundown a battery.

 

One time I accidently left my truck unlocked. One day later my 12v is now 6v...
Door Ajar may be a worse situation because the car may think door is open and controllers don't go to sleep anymore.

 

So what is the solution for a Door Open light? I taped down the rubber button switch tightly with duct tape and the light still stays on. It's impossible to pull out the switch fully because the wire is too tight/short. I will need to disconnect the battery terminal in the trunk every time I park the car to avoid a dead battery!

 

Check the wiring in the rubber boot in the door also. It pries out easily with a screwdriver. I just replaced my left rear door and had to pull it out. I actually have the door wiring harness for sale if needed. Check out the pcs that someone posted in this thread:
Door Ajar Light Stays On - 2016 Sonata Hybrid Limited | Hyundai Forums (hyundai-forums.com
Here's a few pics of the extra harness I have:

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If it's just the switch, you can replace it for cheap too:
(3) Hyundai Sonata (2005-2014) - How to replace Door Jamb Switch |#doorjambswitch #hyundaisonata - YouTube