In terms of charging your laptop's battery and damaging a relevant piece of hardware, tripping over the charging cable from the brick to the laptop is the Number 1 cause. There are three frustratingly weak points along that same section of the charging cable going from the brick to the laptop.
I bet you dollars to doughnuts that at least one of the following was damaged when you tripped: the connection between the cable and the brick, the small section of cable leading into the male plug which you insert into the side or back of your laptop, the copper pin inside the plug's sheath, or the actual receptacle on the laptop's motherboard into which you would normally insert the charging cable's plug.
The last possibility listed is the most expensive to fix, because the piece in question is soldered onto the motherboard and has to be completely replaced with an entirely new housing and guts if any part of it is even ever so slightly damaged. This part costs $90 and up for computer repair stores large enough to negotiate bulk discounts with the manufacturer. Once you have the part, if you are willing to shell out the cash, you will need to almost completely disassemble the laptop's insides just to get access to the part in question. It is no simple task, because you'll be working your way into the nitty gritty guts from underneath the keyboard and from the underside of the laptop. Once you have removed the old, damaged piece, you will need a quality soldering iron, extremely steady hands, and no small amount of skill in soldering with silicon wafers.
A red "X" through the normally pure white battery icon in the system tray of pretty much all versions of Windows means that the laptop's main battery is not present, or if it is seated in the laptop it is not accepting/holding a charge. This almost always means the battery's spirit has already departed for the Great Bit Bucket in the sky, though in this case it could actually be a fault in the charging cables and brick.
In the interest of clarity, please note that your laptop should operate perfectly fine without a battery inserted, so long as you have the charging assembly all ship-shape and plugged into a grounded outlet. In this case the laptop is instead drawing all current directly from the AC outlet. This is why, despite tripping on the cable and having a battery that might well be dead, everything is still powered and all lights are green and good to go. Given all of the above, what you really need are a spare charger and battery which you know to be in good working order so that you can actually begin the troubleshooting process.
Finally, to piece it all together, I have seen and dealt with this exact same problem innumerable times, and through learning the hard way I have found that some laptop chargers can sustain damage, yet still power the laptop, but not charge the battery to prevent the mucked up charger from damaging it. The current directed to charging the battery does not pass through the internal power supply because the battery is normally protected by the charging brick, which converts the incoming current from AC to DC, ups the voltage and amps, and acts as a line conditioner/surge protector. I hope something in this answer helps you find and fix the problem. If only I could inspect it personally, I could tell you right off the bat what major malfunction you're facing.
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