This is depressing me. If you couldn't tell before, I have a very low opinion of Dell. That happened several years ago when I bought a laptop from them and it died after ten months. They said the hard drive failed and sent me a new one. A week or two later, the new one failed. This went on for more than a half dozen hard drives. Then Dell started refusing to honor my warranty. This same thing happened to multiple people I know. My opinion of Dell isn't bad because their products failed, it isn't bad because they couldn't send me a working hard drive, it isn't bad because they would only send me refurbished crap. My opinion of Dell is very bad because they refused to honor my warranty and those of multiple people I know. I wish I were as smart back then as I am now. I learned while looking into your problem that Dell laptops have a historic tendency to have defective motherboards. Defective motherboards that are defective in such a way as to make it appear the hard drive is bad. I think I know what was
really wrong with all of our laptops.
Yeah, that all sounds probable. Based on what I remember, the problem is a pin breaking just before the connector that you actually plug in, that prevents a chip in the adapter from sending a signal that indicates a genuine Dell adapter. Having a difficult time finding references now, though, but given that what I need to do is twist the cord around that spot until it recognizes it, it seems probable that this is the solution.
Looks like Dell itself is completely without liability here, though, given that the purchase was made through a third party seller and that the original cord is far, far beyond warranty.
With the fiery cable, as near as I can tell, it is long since replaced. It doesn't appear to have been involved with any recalls, though. The Dell replacement parts website shows four AC adapters as replacements for this model laptop. Two are 65W and neither is the same as the one pictured. So it's possible it really is the original adapter that came with the thing and Dell has long since replaced it. This laptop came out in early 2008 and was discontinued by early 2009. The Amazon page doesn't give any relevant information on the adapter, so I'm going straight off of visual comparison. The only spec it gives is the power output.
So I got creative. I found adapters that visually match that one that also claim to be for the Inspiron 1525 on eBay. eBay is giving me surprisingly useful specifications. All across the board, different sellers of this adapter all give the same specs. Dell won't tell me anything about the adapters they're selling except that they're 65W or 90W and the only information I can find regarding the power specs of the machine are that the power adapter takes 120-260V and 50-60H. The eBay adapters are consistent with these specs, so in theory should work. I'm not finding anything that says this adapter has a problem with exploding under normal circumstances, so it probably isn't a problem inherent to the adapter. It's either a compatibility issue or, I think the more likely scenario, the thing was defective before igniting.
For your old one that you're now using, if it's working well enough, you're good. I'd advise against trying to fix it. Soldering these sorts of things tends to be sketchy.
On the plus side, you can get this adapter on eBay for <$10. There's one listed at $0.99 right now. The cheapest one Dell is selling goes for $59.99.