You can do a swap if you wish. And it should work, because you will be essentially changing the whole phone sans the case and ancillary components. But remember the TouchID won’t work. And I heard that some second boards could have other problems you cannot expect. I wouldn’t risk it.
This is a 10+ year old laptop which doesn’t support MacOS Sierra or newer. Even El Capitan is not necessary supported. Most likely you would be stuck to Lion or even perhaps a Snow Leopard if you are dealing with the first gen A1181. You are getting no entry sign, as the laptop tries to boot into MacOS it is not supported on this machine. Internet recovery was not a thing on these laptops, and only Late 2009/ Mid 2010 and newer support it. So no Cmd+R or any other recovery methods besides the physical installation media. Your best bet is to get a Snow Leopard Installation DVD, or a Lion Installation USB Stick (this one could be created from the working machine, but you must also obtain the installation image). Or use a Recovery DVDs that the laptop came with. Then you can boot to installer, format the drive and do a fresh install of the OS X.
Usually the AC adapter fails on these machines. In fact, I haven’t seen an AC adapter in the last 5-7 years from these series PowerBooks that was still good. But luckily, they are easily repairable, just need a complete replacement of capacitors. I repaired two in last year and they now work as good as new. So, replace or repair your AC adapter (7.5V 2A) and see if anything changes. If you have a voltmeter nearby, check the output voltage on the adapter output jack.
Was your logic board recapped properly? If not, then do that first, and also clean the PCB very throughly before you solder the new capacitors. They used a type of capacitors on these machines that are prone to leaking and shorting out the logic board after 15-25 years or so. Otherwise, it is also very likely a logic board problem, more exactly it could be a Bourns filter problem. So that needs to be replaced. As the Larry Pina in the book The Dead Mac Scrolls says: "Symptoms: On startup, the desktop appears, but neither thekeyboard nor the mouse work. Typical history: The problem occurred when you connected ordisconnected an ADB device while the computer was on. Probable diagnosis: The problem is on the Mac SE logic board. Solution: Check/replace the shorted fiIter (Apple part 155-0007-E) at board location A 11"
If you follow the official service manual that can be found with some googling, then why not. I’ve done it a few times myself on these machines with no problem. The only major risk is breaking the CRT on the neck itself if you don’t remove the neck board (just pulls straight). Most likely, the floppy drive itself is not broken, just in need of some overdue service, like some lubrication and head cleaning.
Few points to add: Was the logic board recapped and cleaned? Did you try and change or reseat the memory? When was the last time the machine worked properly? Last week? Or 10 years ago? CMOS battery (actually it is called the PRAM battery on these Macs) would not cause much problems on these machines if it goes flat, however it might leak after all that time and destroy the logic board…
If the disks are known good and you can boot from then on an other Mac 128K, then there is most likely something wrong with the drive or perhaps something on logic board. However, the RAM problems usually result in sad Mac with the error code. Usually there is a problem with floppy drive, as the mechanism gets stuck as the old grease hardens and the head assembly would then have a problem advancing on its rails, so it would only read track 0 properly. It is a main part of a Classic Mac restoration/repair to disassemble/clean/lubricate these old drives. I did have problems on a 128K with the boot disks prepared from disk images on a 1998 PowerBook G3 Wallstreet. It just didn’t like them, but disks that were prepared on a Mac SE/30 worked perfectly. Also I needed to use actual 800K disks, not the 1.4M ones.
As far I am concerned, these keyboards are unrepairable when they fail. Well at least you can still change the batteries in those models, the newer ones have a soldered lithium battery. Check this guide, it might provide some help: Apple Wireless Keyboard (A1255) Teardown