I'm not familiar with the double beep. Based on what you've described it sounds like there may be some damage to your phone's charging/data port, which after a fall wouldn't be out of the realm of possibility. Again based on what you've said this is my guess.
When you say that worked initially but stopped working, that says to me that something changed. If not a setting then a break, like something inside of the device broke. Add to that the beeping. Which sounds an awful lot like the device is going in and out of charging. Similar to if you were plugging and unplugging the phone to a charger.
I would carefully plug in the USB plugged into the PC I like this better [Since you know that you can get an image on the TV I might start there rather than on the PC since you'll have a visual rather than a audible feedback and there likely won't be a delay because with charging the indication is not always instantaneous. The phone has to recognize the presence of power then alert to it and a visual output will likely be nearly instant.] and gently move the plug around in the port, I say gently because if it is to the point that it's beeping like it is being plugged and unplugged there is already damage and you don't want to risk damaging it any more. So gently with the phone on it's back lift up the bottom of the device by the cord, hold it in the position with only its own weight pressing on it, and see if the beep stops. If the beep stops try to access the device through the PC and get everything you can as quickly as you can because there's no telling how long the connection might be good for. If the weight of the device is enough you might get away with putting a pen or pencil under the cord to support it in this position. If lifting up doesn't stop the beeps then turn the phone over and try the same thing from the back side and again if the weight of the phone is enough a pen or pencil. Now if neither of those work you could try side to side but things don't usually break that way but as long as we're talking about only using the weight of the phone why not try.
If after checking all four directions and not getting a good result you'll have to step it up a little. I don't know what you have laying around your place so I'll use things that everyone might have or could easily approximate. I'd probably put a pen under the plug and sit a half can of soda far away from the plug. If you know fulcrums you'll know that this will generate the least amount of force. Then slow work it closer to the plug until you get a result or don't get one. Again if face up doesn't work turn it over and do the same from the back side. Since it seems to be making at least some contact, if I'm assessing the conditions correctly, it really shouldn't need a lot of force to reach a point where things are making contact so I wouldn't jump to a full can of soda or any more weight then a gentle press. This is especially true since we don't know which direction creates the connection and you wouldn't want to cause more damage by forcing it in the wrong direction.
Once you've learned the direction and pressure needed you could then transfer what you've learned to making the connection at the PC. For instance, half a deck of cards under the wire at the end of the plug with $3 in quarters in the middle of the phone gets you a picture on the TV, then a similar set-up will hopefully get you the same result on the PC.
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