It's ambiguous, but I would err on the side of yes.

The only justification in the text to allow the illusion to play music is this part:

behave in whatever way you choose.

Which is not super convincing. However, here's my argument for why it should be able to play:

• Your double probably has the same equipment as you: If you were holding a sword or other item, it seems unlikely that the double you create wouldn't have it. Otherwise, one could argue that it appears naked! This means that it likely has the instrument, and thus could play it.
• Creating sounds is level-appropriate: Consider that Mislead is 5th level, whereas Major Image, which can replicate this scenario, is only 3rd level. Moreover, as far as illusion spells go, speaking isn't explicitly different from any other sound. Even the cantrip Minor Illusion gives wide latitude on sound.
• It doesn't break the game: You can't cast spells or anything using this music--the only benefit you could derive from being able to play music would be hiding, which is well within the spirit of the spell. Even if a DM rules that you can't play your instrument, you could simply get around this by singing, anyway.

Illusion spells are not tangible in general.

The mislead spell creates an "illusory double," so we can look at the description of the Illusion school of magic for more insight (see Casting a Spell in the SRD, emphasis mine):

Illusion spells deceive the senses or minds of others. They cause people to see things that are not there, to miss things that are there, to hear phantom noises, or to remember things that never happened.

Since the illusion created by mislead is not really there, it has no physical presence. It could not be touched or bumped into, so it is literally not tangible.

Some illusion spells might create illusions that are tangible, but if they do then we would expect them to explicitly state that they are exceptions to their school of magic. Since mislead does not state any exception, we have no reason to think that it is one. In addition, if it was possible to physically interact with the illusory double, we would expect that the designers would call it an object or a creature, but in the absence of either of those words we must simply assume it to be an effect that behaves exactly as described.

(This is under the interpretation of the word "tangible" meaning "material or substantial." That is, tangibility is an inherent property of a thing regardless of who is or is not perceiving it.)