When the battery is dead this will cause the key to stick in the ignition like this here on this model.Once the battery issues is resolved this will correct the key staying in the ignition fault.
Here is the battery draw test.Keep in mind this test will show all o.k. until the vehicle is acting up which can be tricky to test to locate the fault in some cases but this is the test that needs done to pinpoint the cause here.
what needs done is to leave the key out of the vehicle with the windows rolled up and make sure everything is off with access to the battery.
Set the meter to amperage or milliamp and disconnect the negative cable from the battery.Put the leads in the multi meter the correct way (negative in negative and positive in the amperage or milliamp slot if equipped).
Connect one cable from the meter to the ground cable that was connected to the battery and the other lead to the ground terminal on the battery,this way the current is flowing through the multi meter.
After around 5 to 10 minutes see what the meter reads for a amperage draw.Normally you want to see less then 50 milliamp or less then .05 amp.
If you see over this amount of a battery draw/drain then your vehicle has an electrical component draining the battery.
From here you would remove one fuse at a time with the meter still in place and observe the reading on the meter until you come across one fuse that allows the draw to read below the 50 milliamp.This is the fuse or circuit that has the component on it that is drawing down the battery.Make sure when you remove one fuse if it is not the issue reinstall it and move to the next waiting a minute between each fuse removed to get an accurate reading.
Once you find the fuse or circuit its on then you would look on a wiring diagram to see what that fuse powers and unplug each of those components one at a time until you locate the component causing the draw then this component would need replaced.
Let me know if you need anything else and feel free to rate my help here.Thanks Pete