You battery may be the problem. If the ups cant take the load of just the monitor then probably your battery is gone. Put a multimeter and monitor the voltage from Before turned off to turned on(offload to onload). If the voltage drops below 11v immediately(onload) then ur battery is dead or not charged at all.
Example data:
[OK Battery]
My batt's standby V is 12.9(doesnt tell anything).
On Charging:13.1v(almost full)
On Load: 12.7v (almost full and if i keep the multimeter attached it drops by 0.01v every few minutes i.e getting slowly discharged. this rate increases for smaller batteries)
[Dead/Discharged battery]
Standby: 12.7
Charging(dead one): 13.4 (if Dead it will show fully charged very fast.)
Charging(discharged one): 12.1
OnLoad(dead/discharged): <10.5 - 11.5
And the 10% charging current part is not accurate in practicality. Most batteries are happily charged below 4% their rating. For eg. in a car the charging current (for 35ah) in nominally 1amp only . And the big-truck-batteries 150ah are normally charged at 4amps (not 15amps are u might think. u dont even get a 10+amp charger easily in market)
I would suggest you to get hold of a new battery. may be take the UPS and the multimeter to a battery wala and pay him like 10 bucks to use one of his batteries to test it. You can use ur car's battery. no need to unplug it,just dont get it too discharged or there will be more of the aforesaid 'probations'
Now, there are two things to test for.
1. On Load: Use a charged battery. atleast half way charged. may be try ur dad's car battery
. If it works ok for some load then everything is fine. Which should happen if the battery is not dead/discharged
2. Charging: Worry abt that later. first get the stuff working offline, i.e without power, with a charged battery. If you are happy with the result then start checking the charging current/time.
Note: a battery's stand-alone voltage (like u said 12.4..12.8.. etc is never accurate) Even a dead battery will usually show 12.5 or so volts stand-alone.
Shock(Leakage current in this case) is because of poor/missing earthing in ur house/socket.
And dont call it shock.. it is just some tingling.
Got another query. I have seen many icecream wallas in their trolleys having a tube lit inside during night. They use a single 35/32 Ah battey without any inverter circuits. How do they accomplish that?
they use miniature ctk inverters. available at a lot of places and as DIY kits. Basically the emergency lights without their plastic bodies.
UPDATE:
I kept the battery charging whole night and its still the same. When I switch off the mains and the UPS shifts to battery power(with the PC on), it just shuts off. Do I need a new battery?
Very Probable. to confirm test using a different battery.
btw.. in that image, what kind of wires are you using to connect the external battery. Dont use the ones used in homes. that is for AC and causes a lot of resistance for DC. Use the thick (made of lot of think wires) that is used in inverters to connect the battery.