adapting 60 LED battery operated string-light to work with wall wart
The way these strings work is multiple leds in parallel depending on a single series resistor and/or the the battery Equivalent Series Resistance. Some simpler strings also depend on the Forward Voltage of the LEDs being higher than the battery pack voltage. A 3.3V @ 20mA led will only pull 15 or so milliamps at 3V.

In this case, there is likely a resistor or diode between the battery pack and the led string. Since these have timers, there is probably a mosfet or transistor and an IC internal to the case.

Simplest case, a 4V power supply should be used. Otherwise a 4.5V should be fine. Open the case to confirm there is a resistor or something.

As for current,  measure the led string with fresh batteries using a multimeter.  This will confirm how much current is being pulled.

As for the broken plug in one, what exactly broke? The string? You can solder, crimp or twist it back together, by removing the clear coat insulation. If it's the power supply, then you can't re purpose it. Check with a multimeter.