The LEDs are probably 3.2 to 4v - the battery usage for this (and many such) products is 4.5v (1.5v x 3 batteries), close enough, a little over-driven. We know this because if they were much lower voltage LEDs, they'd burn out wired the way they are in parallel - each LED received 4.5v from the batteries, with the power being distributed to all LEDs via the metal strip inside the curvy plastic channels. If they were much higher voltage, they wouldn't light up with only 4.5v.
Because batteries are inherently current limited, they will drain faster, but the current will not run uncontrolled and melt anything down. So if you want to power this from an external power supply (transformer, etc.), you have to limit the current, either with a current limited voltage converter or with resistors (with or without a voltage converter, depending on the supply and resistors used).
When working on this thing you could re-use some of its solder points, but in the illustration below I've avoided doing that for one important reason: making sure you don't melt any plastic.
The green circles indicate an easy place to cut the supply wires and the grey circles indicate where you can solder some additional wires which will then terminate at a connector for hooking up your power supply/transformer. You can use a bullet style DC-connector, a molex, mini-J, etc… Whatever is convenient for you. For supply you can use or re-use a wall wart from something else, buy something specific for this, etc.
Every white circle indicates one of the original LEDs - you don't have to do anything with these, but if you wanted to change out the color temperature or put in different colors, you could melt the hot glue and pull them out.
The yellow-orange rectangles indicate where you can cut the positive wire and attach, in-line a resistor. The value of the resistor will depend on the desired power supply you're going to use. For my own displays, because I was already using 12v for everything else, I went with that and tied my Tumbler into the main supply. For my Cloud City weather vane, because I didn't want to crack it open any more than I already had, I used a current limiting voltage transformer to get the correct voltage and not burn anything out.
You can *probably* assume these LEDs are 3.4v and 20-25mA. *Over-spec the resistors a little bit to be on the conservative side.
Consult this easy reference at Instructables to figure out what resistor values you will need, depending on your supply voltage:
https://www.instructables.com/id/Choosing-The-Resistor-To-Use-With-LEDs/
Or buy a current limiting DC to DC voltage converter on eBay to use with your desired power supply and omit the resistors.