Part of it is the age, the battery will only last about 2 years. The Tesla Electric Sports car has 10,000 computer battery in long tubes and every two years you have to replace 10,000 batteries because they are spent.
All current designed battery die, from being charged and discharged, the heat from the expansion and contraction .. breaks them down into dust eventually.
The other part is that you dropped it with out a protective case on apparently.
That might have broken the electronics in it, apparently so, as when the iPhone is plugged in to the wall charger or the iMac USB system, there would be the power for it to work ... if it was functional.
The battery is there, to power it up .. when the iPhone is not plugged in.
So ... easy answer ... is trash your iPhone 4 .. go recycle it.
Hard answer is take your iPhone to a repair shop and have it refurbished which is a waste of money as it can not take the new software.
The iPhone 5s is $149 ... under contract for two years of course ... but you get a new one that will work for you, take the new software updates, and last for a while ...
until you drop it again and destroy it.
To fix that .. you have to get Traditional 5 element Chinese Acupuncture.
Your Defense Structure is trying to make you cry out the bitter salty tears and fix up your broken heart.
Do it not .. then you will just break cars, girl friends, marriages, jobs ... etc.
When you hate your dad you will have divorces, affairs and dead children.
When you hate your mom you will be an addict.
Love is what heals ... and you did not get much of that in your childhood.
Your mom and dad are broken .. they are stuck in their Toddler Tantrum mentality from their own poor childhood.
They raised you .. and that broken-ness got into your Duck Disease as part of the Human Condition.
Every one has that problem and every one has to fix it up.
Here is a video on it .. if you want to learn more about it.
https://www.youtube.com/watch?v=4Z8aotZycx0