Hi. I'm no expert on taking things apart -- boomboxes or otherwise. (That Aiwa CS-600U misadventure proves as much. :annoyed
But member baddboybill is one of the earlier people to sign up here on Boomboxery. I think he has a heap of boombox-repair experience. His "electronic cleaner" suggestion might be helpful.
Another member like superduper seems quite knowledgeable, too. Maybe he will chime in. (Shrug)
Yes, it can be said that headphone usage is quite risky to one's hearing, perhaps especially after age twenty. (While speaker usage can be quite risky to the peace and quiet in that home or office SMH)
But I suggest that you keep trying to gain speaker output as the boombox was designed to (quite boisterously) offer. Hopefully nobody will need to de-solder wires and re-solder them to parts of the integrated circuit (IC) board those coloured strands obviously weren't designed to be directly hooked up to -- even if such a (literal) hack does seem to work. (Uh, what Sharp model is it?)
Not sure how to take this post. Its phrasing seems to indicate that I don't know what I am talking about. However, sometimes it is difficult to determine context and tone in writing. So....just to avoid any confusion, I would like to make clear that I was attempting to help by adding to others recommendations and experience with my own. This was not intended to debunk anyone's idea of what the issue could be. It very well could be dirty and could be resolved with a spray of contact cleaner. That's true. A loose or cracked solder joint can cause this as well. I've seen it.
I agree that baddboybill and superduper are very knowledgeable members and have been on this forum for many years. I would never second guess their knowledge and experience no matter how long they had been a member here. They are on top of their game when it comes to repairing these things. However, I'm sure they'd be just as smart if they had just joined so we can't determine ones repair diagnostic and troubleshooting skills on length of membership. If that were the case I have you beat by 5 months

but that doesn't mean I am more or less knowledgeable than you or anyone else. I do tend to offer my assistance when I feel I may have a valid point or suggestion. If I don't know, I remain silent and learn from others experiences on the forums.
Secondly, I never said to "de-solder wires and re-solder them to parts of the integrated circuit (IC) board" that "obviously weren't designed to be directly hooked up to". I said that depending on the design of the box, you may need to re-flow the solder joints to the headphone jack on the printed circuit board (PCB) it's attached to.
I am sure superduper and baddboybill have both seen this happen because I most certainly have.