If you are ordering parts, you might consider getting some additional stuff too - New compression gasket for the removable baffle will be needed, you can order the correct gasket from Falcon Acoustics in England or via their ebay account. Most common US draft seal gaskets are too thick, too stiff (not compressible enough) and most too wide for this application. If the speaker was ever opened up, look carefully at the 8 baffle screws - they are countersunk flat head phillips wood screws, usually brass, #6 or #8, by about 1 inch I think (From memory). if they are at all head stripped, they should be replaced. Importantly, use a properly sized philips screwdriver and work carefully, using your fingers to shield the drivers from any misadventure with the screwdriver. If you're really steady, you can use a power driver to remove the screws. NOTE: when you tighten these baffle screws, they screw into a set of soft beech battens (supports) in the cabinet and are very prone to stripping the threads in the wood so tighten carefully and evenly. if you do overtighten and strip the thread, you'll need to use a slightly larger screw, or insert a toothpick into the hole to give the original screw some bite. Do not use any kind of thread lock or glue.
Examine the baffle for any parts of the velcro that are not stapled down well. Once all the screws have been taken out of the baffle, place a towel on the floor (or do this on carpet). Place the speaker baffle down and tap the front of the speaker onto the carpet softly yet firmly. What you are trying to do is break the air seal (which may be easy with the loose connectors) and get the baffle to start coming loose from the cabinet. If it does, gently lift the baffle/drivers out but be careful, there is limited length of wire from the terminals to the crossover mounted on the baffle. In one speaker this will be important, in the one needing repair, likely both terminals will come out fairly easily and make separating the baffle from the cabinet easy.
At this point, take a few pictures of the drivers, crossover and back of baffle, noting and documenting the woofer driver date codes, condition of wires and solder joints, and condition of the crossover components. The back of the baffle should have the remnants of the old compressed foam gasket. If you have the new gasket, you can scrape off the old from the baffle and battens, and apply the new gasket to the battens. Also note on many Rogers LS3/5a's, there will be a hand written number on the back bottom of the baffle which should correspond to the serial number and A/B assignment. This was part of their QC. If you like, you can also examine the cabinet for a date stamp and sticker from Greaves and Sons (cabinetmakers) - it is sometimes under the white or yellow acoustic foam lining. Cross check the cabinet date against the woofer date code, they should usually be within a few months of each other. If any of the crossover resistors have been damaged or burned, or there are burn marks on the side foams, then we'll need a pic to check it out.
Now you can get to the repair. Now you can see why they initially chose the threaded collar fastening - the jack can be soldered on easily, then fed from inside out, and then tightened. With most other binding posts, they must be inserted from behind, then locked down, soldered, and then the speaker reassembled. The tight confines of the cabinet make it difficult to do this but it can be done. Don't be surprised to burn a little foam while soldering, happens. A soldered ring tab which can be fastened to the threaded post is a lot easier to lock down with a lock washer and nut. The 5 way binding post then allows you to use bare wire, thin pins, spades, banana plugs, or other wire attachments. The other speaker can be similarly updated. Be sure to maintain proper color coordination with the crossover wires.
When all the soldering is done, do a once over light tightening with your screwdriver for the tweeter screws, nut driver and screwdriver for the crossover screws. For the woofer, a light nut tightening is all that is needed, the screws are special double threaded so do not attempt to tighten or loosen the woofer screws from the front.
Assuming no woofer cone or tweeter dome damage (if the metal perforated tweeter grille is still present, should be ok), then you can return the baffle to the cabinet, evenly placing the baffle down on the new gasket. Then thread all 8 screws and cross tighten till all hand tight. As mentioned before, it is not useful to overtighten so a good hand tightening is plenty. Then push on the woofer cone and see if it easily springs back (bad) or comes back slowly (good). This is 'the box test' to see that the system is well sealed. Any air leak will mistune/mis-load the system and must be corrected. There is a woofer frame seal that assists with that, but is usually not renewed unless the driver is removed or replaced. Do NOT use any type of silicone, blu-tack, or other sealant - you would have a very tough time ever taking the baffle off again.
If you have the baffles out, that's a good time to do any cabinet cleanup you need. There are various methods and materials depending on veneer finish and extent of damage or problems.
Once you have completed your repair, you can place on stands 24-32 inches high depending on your listening position. make sure your speaker wires are not so thick and heavy as to pull the speakers off the stands. Place usually at least 12-20 inches away from back and side walls, and about 6 feet or more from listening position. Play a mono signal to ensure a solid center image and bass response, this will confirm proper polarity from system and speakers. if not centered or bass output weak, polarity is reversed somewhere in the system. Toe in as needed for imaging but start with facing straight forward.
Remember, these are small speakers and power limited - they are 15 ohm so take more voltage than typical to run so 25 to 75 watts of clean power is needed. But do not try to overdrive for high volume or big bass, they won't like that and may bottom and damage the woofers. They do work well with tube amps. At that point you can let us know what you think of the speakers.