I'm a little confused about why you are talking about the LEDs. The LEDs show the status of power and such but have nothing to do with turning power onto the device. The power button is the only thing you should need to be concerned with. The power button appears to be above the keyboard but I could see them running the wires through the keyboard. You suggest that if you plug the keyboard in then the power works. So if you can trace the power button back to which of the wires on that connector they go to, you can find out which wires are needed to turn the system on/off. But being a laptop, they like to use these small plastic ribbon cables which maybe there are some standard to but they often seem to be very specific for the one laptop they were made for. Not sure if you can easily find something that would plug in here and give you a more normal pair of wires you can connect to. I can't say I've ever seen anything like that. If you're careful, you could probably solder some small wires to the pins on the connector itself and skip the ribbon cable. Or maybe you could cut out all of the ribbon cable except the part that goes to the power button.
I have a whole stack of old laptops I have plans for doing something with. I like the idea of using old rack mount network equipment and such and mounting the parts in there so I can mount them more easily than a laptop normally would and use them for random things. I've tested many of them working on my desk by connecting up the needed parts. But most of them have a separate connector for the button, some with the LEDs on the same board as the power button. I only ran into one old Dell that was stupid and had a hardware lock that would prevent it from booting unless the display itself was connected. Not sure why they would do something stupid like that. But I got around that by disconnecting the display controller from the display panel and that was enough to let it boot without the whole display. But I think it would still only show the boot screens to the no removed display and only after booting into Windows would an external display turn on. It was pretty crap so I don't think that one ended up in the pile of parts for disposal. I have plenty of others that are more interesting.
The E210882 sounds like the motherboard model number. What's the PC model? They might have used the same motherboard in different cases. But from what I'm seeing, the motherboard looks to be a microATX form factor so any microATX motherboard should mount in there. The rear panel might be built into the case or at least not a standard size. So you might have to cut that open to fit a normal rear panel port shield. I would assume you'd swap out the PSU and not try to use the old Dell one so you shouldn't have to worry about any PSU compatibility. The only other thing would be the front panel connector. Dell generally documents their PCs pretty well. So if you look up the model number or Service Tag, there's a good chance they list what pins are for power button, LED and so on. Maybe if you do some searching you "might" find an adapter. But probably you'll have to make your own adapter or splice in some normal dupont connectors.