Reddit mentions of Excelvan 1M/3.3FT IP65 Waterproof Flexible Color Changing RGB SMD 5050 30 LEDs DC 5V Battery-powered LED Strip Light with Mini Controller

I used this LED strip. While it's a strip rather than a rope, it does have adhesive on the back of it that you could stick to a wire or something to keep its shape.

A bonus is the variety of colors and modes. Here's an album showing how I just implemented it last night linky

reddiquette I made one of these a while back, here's how I made it. Obviously feel free to the substitute the materials I listed for whatever you want. The final product (I'll post pics later) wasn't exactly beautiful, but it works!

Materials:

-Some type of container that will make up the structure of your infinity mirror. I used a paper mache box that is sturdy and able to easily open up


-A mirror. This should be a similar size to the container above.


-Adhesive film with a transparent side and a mirror side


-Plexiglass

-Something to cut the plexiglass with

-Some sweet lights


-Glue

Steps:
(for the purpose of keeping it simple I'm going to assume the materials are the same ones I used that are listed above).

1. Cut out the bottom of the box (the part that doesn't lift up) so that the whole mirror (minus the edges) is visible when the box is open.
   
2. Glue the edges of the mirror to the edges of the box so that is firmly in place
   
3. Cut the plexiglass with the special plexiglass cutting tool so that it is exactly the same length and width as the top face of box. This step might be a little difficult as the plexiglass is kinda thick. I found that cutting it from either side and then 'snapping' it produced a nice clean break
   
4. Cut the adhesive film to the same length and width of the piece of plexigalss you just cut
   
5. On ONE side of the plexiglass, stick on the adhesive film so that it fully covers one side. Try and make it as smooth as possible without any air bubbles/wrinkles - think of it as putting on a screen protector on a phone
   
6. Cut a hole in the top of the box leaving around an inch on each side
   
7. Open the box up and using the glue, stick the piece of plexiglass to the underside of the top face using the inch margins on each side. NOT THAT THE FILM MIRROR SIDE OF THE PLEXIGLASS SHOULD FACE INWARDS TOWARD THE BOX)
   
8. With the box still open, stick the LED lights around the inside perimeter of the side of the box. There may be some excess length which is fine
   
9. Cut a hole in the side of the box large enough for any excess lights and the controller to fit through and then pull them thru so that they are on the outside of the box.
   
10. Close the box up and turn on the lights! The ones I recommended have a ton of different color, flashing and speed settings so you can create some pretty cool effects.
    
    
    I know that at first glance this might look pretty confusing. As soon as I get some time I'll post some pics of each stage to help with the explanations, as even reading back on them they do sound somewhat confusing!
    

$11 on Amazon

battery powered LED strip takes care of that part of your question. The "go with music" part sounds like a job for a Arduino or Raspberry Pi project.