Yeah the LEDs are low quality, tiny die type that can’t handle the drive current. While I agree that doubling or tripling the resistor value would fix it, I think that might be too dim for the effect you want.
Instead I would consider what materials and components you have on hand. If you have a spare scrap of perfboard or other PCB material, or (lol) even cardboard, you could abandon that circuit board and wire up one resistor per LED, using around 100ohm/LED, or one resistor for each set of two in parallel with 47 ohm /2 LEDs.
If you just want to do it the cheap quick way, you could just drill holes for the leads and add three more LEDs to that circuit board, scratching away copper traces where needed and wire-wrap connecting them in parallel to those already there, then you have half the drive current for each LED and about the same (slightly higher) brightness. However I hate to see so many LEDs in parallel as their minor forward voltage difference will mean some die earlier than others then cascade failure of each weakest one in turn, which is why I initially suggested that each have its own resistor.
How long do you want it to last? Whatever # of LEDs and drive current and # of resistors, try to aim for 10mA at most to get good LONG life out of them. 20mA just isn’t a realistic spec for many generic LEDs I’ve tried, even with ample copper on the PCB they still degrade from constant use.
Today higher power LEDs are cheap, so I’d as soon under-drive higher powered LEDs. You’re using white so I’d pick up a 5 pack of XT-E from FastTech for $3.87:
https://www.fasttech.com/product/2058900-cree-xt-e-450lm-4500-5000k-led-emitter-5-pack
Drive ONE of those XT-E with that 10 ohm resistor in series and it’s going to be about 130mA peak drive current off a fresh charged battery, a little under half a watt so you don’t need any heatsink, just mount the star directly to the area where the existing PCB is/was. Plus, then you have 4 other XT-E to play with… they make a sickly greenish-white tinted light, but it’s not terrible if you don’t have something else to compare against, and they are a very cheap way to light up things where you just want up to 100 lumens or so with the bare minimum addt’l parts possible.
However, you may not need that many lumens, so if you happened to have a 22ohm, 1/4W resistor lying around, that’s what I’d use for this project. Lol, that’s close to what Barkuti already suggested to keep using the same LEDs, but I think it is pushing it to expect only 3 of those 5mm single-die LEDs to provide enough light with good lifespan.