It seems to me (from an abbreviated check of "how to fix that type of keyboard" images) that you've got a fairly common "scissor/pantograph"-over-rub er-dome-switch keyboard (probably not the 'official' name). It uses membrane-like contact switches below a mechanically independant keytop to have some of the advantages of a pure-mechanical key-switch (the 'feel') and membrane (the thin nature), with the (part-spring part-poke-the-membrane) domed-rubber sheet making tactile contact between the two.
Photos also show a metal backplate, that should have prevented the membrane layers from shifting seperating easily, and top-down damage would more obviously result in loss (or looseness) of the keytops. (Not many views that give any decent edge-on perspective, but seems fairly likely to be thin (aluminium?) baseplate, two or three 'contact' layers, rubber-bubbles poking up through the plastic 'foreplate' upon which all the pantographs clip, and the keycaps onto them, probably with strategic nubs from the rear of the plastic plate, through incisions in the membranes, to key in/attach through the rear metal plate and prevent latreal dislocation of components, also accidental[1] disdassembly.) The connector looks too 'thin' to have separate column-of-keys fully parallelised connector, so it might be serialised down within a sub-board within the structure of the keyboard unit.
You don't need to worry about the 'live' power going through the keyboard's power-switch, it'll be a low-voltage connector signal at that stage, and prompt the mobo/PSU sub-board's power-switching function. But I'm less sure that you'd easily identify the physical problems that I'm anticipating you'll have to uncover, so maybe it's not something I'd ask you to do yourself. (Also, according to one "replace a T410 keyboard" walkthrough, you need to (re?)remove all kinds of things, like the RAM, to get to the keyboard screw. No doubt you did this all, 'willingly', to replace the mobo itself, but you'd want to avoid doing it all again until actually necessary, and running it 'half re-opened, half re-reinstalled' might be outside your comfort zone.)
If you were anywhere near me/some computer workshops I know well, I would prefer to suggest you just pop it over so me or my occasional 'colleagues' would give it a quick once-over and then tell you if it's a big problem or (equally possible) so minor that it was fixed inbetween unscrewing everything and screwing it all back up, for zero/peppergrain total cost. Perhaps you can still find a sympathetic (and honest) workshop by yourself, perhaps glad for once that it's not as difficult to repair (or even open) as a water-damaged iPhone. Yet, notwithstanding anything LB might say (and I'm a bit rusty at handling random 'pop-up' repairs, so possibly LB rightfully has more optimism), I'm more and more tempted to suggest getting a brand new KB module (not cheap, of course, but not necessarily expensive either). Though still some chance of fudging it back into shape, with appropriate amounts of luck and care. I'd be happier with my own hands on it, but ...what else can we suggest that you do? I'm resorting to just explaining what my current full understanding is, but I'm not sure how useful it's being. (I know I operate mostly with mental images and concepts, but not always so good at verbalising them.)
...so. Where are we again?
[1] Or deliberate... I mean, you probably can separate it if you quickly need to dry it out from a drinks-spill on it, but they often try to make them into no-user-servicable-parts as a unit component. Because separating the electrical-contact layers can cause problems of the kind you're getting, though rarely just that, if care isn't taken to reunite them.