I had a quick look through the service manual for the TK-260 and it seems that those radios beep and flash for a number of reasons, though most appear to be unrelated to programming.
One that did stand out because it agreed 100% with the description you gave was that if the battery is very low the unit will flash a red LED and emit a tone. I know it's a silly thing to ask but could that be what the problem is?
Also, again asking the obvious, have you tried doing a full reset on the radio and then trying programming again?
If you have another handheld available have you tried checking the cable/software combo on that (to see if it's the programming items that are causing the problem?)
If it's possible to clone rather than program the radio, have you tried cloning from another set to see if that cures the problem?
On radios that use the mic/earphone sockets as the connection point for programming leads it's worth making sure that the socket is clean and free from corrosion, and also that the socket hasn't become broken away from the PCB, as this can lead to a loss of data during programming and render the unit 'brain dead'.
Having said all of the above I have had a number of programmable items 'die' for no apparent reason over the years.
All part of the 'mystery' of Amateur Radio I guess!
73, Mark.