Exactly what I was thinking!
Of course, when you daisy chain them together, the very first and very last connected candles will have the 'open' socket to feed in power. If you are fine with this proceed! Otherwise, you could stick a third female socket in a 'hub' candle which would allow you to put power in anywhere in the middle of the string of candles. (Wherever the 'hub' is)
Now... as far as the barrel plug is concerned, they are often center positive, but they can be wired with the center negative too!
Take a look on the pack that plugs in the wall like Davenn suggested, you should find the following picture on it:
View attachment 18871
If this image does not exist, you will need to determine this yourself... Ideally, you would use a multi-meter or volt-meter to test. Black probe on the outside and red on the inside. If you get a positive reading, you have the left image, if you get no reading or a negative reading you got the right image. (Swap the red and black probes to be sure)
As far as wiring is concerned, you can find all sorts of tables online telling you what gauge wire you need to handle the amount of current you will be providing to the lights.
Personally, I use the table from :
http://www.powerstream.com/Wire_Size.htm
So... what does that mean exactly? Well... if the wire is too thin, or too long the voltage will drop in the wire! So the 4.5V AC-DC adaptor you have may only provide 3.5V which I'm certain would be noticeable and undesired.
Personally, I would try to use 18 gauge wire (But only if I can comfortable attach this to the female barrel plugs.) I've chosen this based on the 'transmission' column... partly because I am unsure how long the patch cords might be in total, but you can choose smaller wire if you have any ideas or limits on the maximum length that your string of candles will be.
*I should clarify the use of the 18 gauge wire. This ideally should be the gauge of the wire from the AC-DC adaptor, the gauge of the wire in the patch cords, and the gauge of the wire that connects the female barrel jacks together internally because this will be the 'backbone' that all the current will flow through. Each candle will 'branch' off from this main line, so the wire that connects the actual candle electronics to the power line you are putting in can be much smaller. (You can use the existing wire that was used for the battery pack if you want to keep is easy... Soldering onto existing circuit boards can sometimes be ... tricky depending on the quality it was originally put together with.)