All rechargeable batteries must be periodically charged to maintain their health - in particular Li-ion batteries used in portable devices such as iPad.
All rechargeable batteries will self-discharge over time. If not properly maintained, a fully discharged Li-ion battery can be damaged as it falls into a state of "deep discharge" from which it cannot recover; if this should occur, the only remedy is replacement of the device battery. Your description strongly suggests that this is the situation with your iPad.
For information, Apple's published advice is as follows:
Store it half-charged when you store it long term.
If you want to store your device long term, two key factors will affect the overall health of your battery: the environmental temperature and the percentage of charge on the battery when it’s powered down for storage. Therefore, we recommend the following:
- Do not fully charge or fully discharge your device’s battery — charge it to around 50%. If you store a device when its battery is fully discharged, the battery could fall into a deep discharge state, which renders it incapable of holding a charge. Conversely, if you store it fully charged for an extended period of time, the battery may lose some capacity, leading to shorter battery life.
- Power down the device to avoid additional battery use.
- Place your device in a cool, moisture-free environment that’s less than 32° C (90° F).
- If you plan to store your device for longer than six months, charge it to 50% every six months.
Depending on how long you store your device, it may be in a low-battery state when you remove it from long-term storage. After it’s removed from storage, it may require 20 minutes of charging with the original adapter before you can use it.
The full text can be found here:
Batteries - Maximizing Performance - Apple
You should note that long-term storage requires the iPad to be properly shut-down. If not actually shut-down, a fully charged iPad battery may be fully depleted by background running processes within a week (or less). While the iPad will eventually automatically shut-down to protect the battery from immediate damage, the battery must be recharged before irreversible damage to the battery occurs. An iPad must never be set aside with a fully discharged battery.
Battery Replacement Service
You can establish eligibility and cost of a battery replacement in your country/region here:
iPad Repair & Service - Apple Support
You might choose to visit the Genius Bar at your local Apple Store or Apple Authorised Service Provider (AASP) - where the technicians will be able to test and assess your iPad - prior to committing to a battery replacement or other repair:
Genius Bar Reservation and Apple Support Options - Apple
For practical purposes, in lieu of servicing and returning your own iPad, Apple usually exchange the iPad for a renewed/refurbished iPad of the same model and specification. The cost of the exchange is the same as would otherwise be charged for battery replacement. A out-of-warranty repair, while expensive, will be somewhat less than the cost of a replacement iPad.
If you have an active AppleCare+ plan (this, perhaps, being the repair package that you declined) the cost of repair by Apple is reduced to a small excess/deductible fee.
