Are you sure of the model number? The D1680 should register the handset when you put it on the base for 30 seconds. (You should see Handset Registering followed by Registration Complete.) Then press Talk/Flash to check for dial tone. On this model if the registration fails, charge the battery (or replace the battery and then charge that). Then repeat the registration.
The manual for the Uniden D1680 is available here: http://news.uniden.com/pdf/D1680om.pdf
For other Uniden phones, like the DEC1060, you need to navigate through the handset menus to find the Handset Registration command. Then press and hold the Register Handset (Find Handset) key on the base for 3+ secondsThen the base and handset will synch. (Low batteries could interfere with the process.)
For the D3288, you access the base menu for Register Accy and then press and hold # on the handset for 30+ seconds. (You will see Registering followed by Registration Complete.)
Additional Uniden cordless phone manuals are available here: http://www.uniden.com/scat/cordlessphones_mnl
I hope this helps.
(with a flea market phone, my first suggestion is to try a new battery for the handset. They only have a given number of recharge cycles and you do not know the history. However, sometimes the chargers do fail and a new battery will not work. In that case, the phone should be replaced. It isn't worth the cost of out-of-warranty repairs.)