modify 700VA UPS to power 65Watt Laptop Adapter for hours instead of minutes
Your UPS is from 2009.  I assume that the battery is also almost 7 years old as well.  This may be the problem.

Lead acid battery 12 volt, 7Ah.  Now UPS usually have a cutoff voltage where they stop draining the battery to prevent damage to the battery.  For lead acid battery, they can be easily damaged when discharged past 50%.  There are more expensive deep cycle lead acid batteries that can be discharged safely to 70-80%, but it is unknown which type your UPS has.

So let's assume that your UPS sets the cutoff point so that after 50% are used, it turns off.  12 volts X 3.5Ah = 42 Wh.  Now add to this the fact that a 7 year old battery in a UPS will have suffered significant capacity loss.  So now your useable energy will be even less than 42 Wh.

Your laptop is rated for 65 watts. So if it ran at the 65 watts for an hour it would require 65 Wh of energy and you have less than 42 Wh.  I'll let you do the rest of the math, but you can see, even if you got 80% discharge, that low capacity battery coupled with its age - you will never get the several hours your looking for. 

Adding to your dilemma, is that the inverter portion of your UPS is not perfect and you will lose at least 15% of your power to it.  So 42 Wh is now down to 36 Wh.

It is possible that the UPS has a timer that shuts down after 15 minutes regardless of remaining power, and you could easily reset this by measuring the voltage of the battery immediately after it shuts down.  If the voltage is around 11-11.5 volts then the battery probably ran out.  If voltage is significantly higher, then a timer shut you down. You will need to measure the voltage right after shutdown, don't wait or the voltage is less reliable.