It takes between 300 and 1000 amps, especially at the start of cranking, to crank over an engine. Lead-acid batteries aren't very good batteries generally, but they are very good at supplying a huge surge of power, and they're cheap. So they are perfect for engine starting.
I didn't have a jumper cable with me, but I had a car battery charger. So, I hooked it up to the battery and tried starting the car. Unfortunately, it didn't seem to give it a jump.
A "jump starter", the kind with huge battery cables, is largely responsible for that 200-1000 amp surge of power. That presumes it can make a very solid connection to the battery. If the battery connection is not solid, it will refuse to start, exactly as you describe.
However, a "battery charger" is designed to provide 5 or 10 amps and provide it for a long time. Using this type of charger is a job for the patient. If it is (mis)used for jump starting, it might provide 30-50 amps until it burns up. They make larger boost chargers capable of more; see the device's rating.
So, to use a normal battery charger this way, put it on for at least an hour to put a few amp-hours into the battery. Really, a battery is typically rated ~100 amp-hours (diminishing as it ages and is abused), so leave it on all day for a decent recharge.
Now if you do all that and it still doesn't start, remember the part where I said lead-acid batteries are lousy but cheap? The "lousy" part is they only have a 4-6 year practical life. The symptoms you describe are totally consistent with a battery at end-of-life. The "cheap" part is they only cost $100-ish new, if you are a sharp shopper. Changing one is literally 4 bolts, however they are heavy as beans.
Even though they're lead which is poisonous, society is really, really good at battery recycling with over 99% of car batteries truly recycled into useful things. In fact the new battery will have a $10-20 core charge that you will get back when you return the old battery.
If you find yourself with an old car battery and no opportunity to collect a core, leave it at any auto parts store -the'll throw it in with the others and collect a few dollars for it from the scrap dealer.