There's no way to force the laptop charge its battery with a charger it doesn't recognizes.
Only solutions to restore full CPU speed.
1. In Device manager uninstall microsoft AC Adaptor – unplug AC jack and then plug back. (helped for some peoples)
2. using ThrottleStop software http://forum.notebookreview.com/threads/the-throttlestop-guide.531329/
Please read carefully the info by the link.
Before making adjustments within ThrottleStop, find out what cpu is in your Dell laptop and figure out what are the normal speed(s) and features of that chip.
Read the 1st post from the above link. According to the author, Dell will either reduce Clock Modulation (CMod) or Chipset Clock Mod (Chip). Download, install, and run ThrottleStop. From the readings, whatever is not 100% as listed on the table is what you need to check before hitting Save. In my case, the CMod reading was 75%. So, I checked Clock Modulation to stop the throttling.
Also make sure the Set Multiplier Number matches your cpu. The default listed was not the proper multiplier for my cpu.
After the above steps, your cpu should now run at the maximum speed. That is great except laptop cpu’s run at multiple speeds to keep things cool and extend battery life. So, you probably want to check Power Saver or SLFM if the chip supports that.
Post 2 from the link will explain how to use the Windows Task Scheduler to start ThrottleStop automatically when booting. Alternatively, you could also put a shortcut in the Startup folder in Windows if you prefer. If you are going to have ThrottleStop boot everytime, you might want to go to Options and check Start Miminized, Minimized on Close, and uncheck Task Bar. These options will make Throttlestop less intrusive so you do not have to do anything everytime you computer starts.
But the issue is in damaged power cord. So maybe just try to locate where the power cord is damaged, cut off the problem area and reconnect the wires directly?
Anyway, nothing to loose already.