If by "a long time" you mean more than a couple of years, then my first guess would be that the lithium Ion battery is totally dead. Li-Ion batteries only have a 2 to 3 year sable lifespan, whether they're used or not. There are sometimes anomalous cases where a battery will work longer, but usually not. Since the device will not work on external power without a working battery installed, it seems your only option is to fin a new battery. There used to be a store called casiobattery.com - might still be there, or perhaps some other supplier, like LionBattery.com...
I also left my E-105 unused for over a year and it wouldn't start. I brought the battery back to life enough to use the device by replacing the CR-2032 battery and tricking the device into trying to charge the exhausted main battery.
I tried the trick (below) mainly to see if the device worked before replacing the main battery. If it had just confirmed the E-105 worked (without also charging the main battery as it did) I next planned to test the charger was working. I never got to that second step because my E-105 is happy as a clam now.
If you are considering trying this yourself, note that this involves working with electricity, so be safe. In particular, I don’t recommend trying this with regular non-rechargeable batteries because the Casio charger will be placing a charge load on them. All that said, the trick is make a set of 3 AA rechargable batteries in series. Place the set (which should produce about 3.6V) in parallel with the main battery for 3 minutes while the external power adaptor is attached. To do this, I ran a short lead from the + and – battery terminals to a jig holding 3 AA rechargeable batteries. I think it worked because the AA rechargeable batteries ran a load across the main batteries without regard for their status, or because the rechargeable batteries made the E-105’s charging system believe the battery was good, or both.
If you’re really feeling more adventurous, you could also consider replacing the E-105 battery back with 3 current technology rechargeable AAA cells. I think they’d fit, but I don’t know what to do about the temperature connector of the OEM battery pack.
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