When the In Use light stays on, there is usually a problem with your telephone wiring. This means that the phone can't hear a dial tone when in standby.
First try a new RJ-11 cable between the phone base and the jack. If you see the In Use light go off, check if you can hear a dial tone. Then the problem was the phone cable.
For the rest of the troubleshooting steps, I prefer to use an old corded phone. Then you don't have to make sure to have 120V power outlets at each test point.
Try connecting your test phone to the original jack where the KXTG-5583 is plugged in. If you get a dial tone here, then the KXTG-5583 is faulty. It has a dial tone but there is a problem with signal transmission/reception between the base and the handsets. (This is actually very unlikely but it's a simple check to make.)
Next, try the test phone at another jack connected to this phone line, if you have one in the living area. If you have a dial tone here, then the wiring in the first jack or between that jack and the service box is faulty. If you do not have a dial tone or do not have another jack on the line, find your telephone service box and open it if you can. If there are RJ-11 ports in the box, connect the test phone to the service line. If you have a dial tone at this point, the problem is within the house wiring. If you do not have a dial tone or can't test for a dial tone, contact your telephone service provider. They will test the line and repair outside problems for free. (This assumes that you aren't in an area where the provider is trying to eliminate their landline services.)
If you found that the problem was inside and only have one jack (or one jack was faulty), you can check the wiring inside the jack. Take off the cover plate and look at the wiring; you are looking for a loose or broken wire. There are two common types of jacks, screw terminals and key stones. For the screw terminal RJ-11 jacks, if you find a loose or broken wire, strip the insulation from the undamaged part of that conductor. Then secure it under the appropriate screw. For keystones, you'll need a punch down tool which is available at most hardware stores in the wiring area. Take off the cover over the wires and locate a loose or broken conductor. Do not strip the wire's insulation, Place the wire above the appropriate point on the keystone and use the punch down tool to push the wire fully into the socket. The wire's insulation will be removed at the necessary spots as it is pushed into place. Replace the cover. In both cases, replace the cover plate (with the keystone if you dislodged it to work on the wires). See http://www.lanshack.com/wire_phone_jack.aspx
for a picture of the wire color to terminal location for the RJ-11 jack.
If the problem is in the wiring inside the walls, you may want to have someone else find the problem. Your phone company will often offer to do the work for a fee.
I hope this helps.
(I have had the phone line go down several times. With copper wire, the problem has always been outside. The cause most of the time was mice chewing on the wires. Twice the problem was due to a contractor cutting the wires while digging near the feed lines. Since changing to a fiber optic system, I've had a single jack go down and sporadically lost all incoming services. The single jack lost the dial tone the day of the switch over. The service technician failed to connect all of the wires to the new service box. I found the wire behind the siding and connected it myself. The complete signal loss was due to a problem with the "electronic brain" of the system and required the service technician to replace the card.)