All that being said, their out-of-warranty options are abysmal. The work they do is fine, but the pricing is completely off the charts and not comprehensible.
A friend just had their cracked T410s LCD replaced and it was $700 (in the U.S.) I doubt an X200 motherboard would be less. Your best bet is a local repair shop, or if you can do the work, to buy a motherboard from a forum member or via eBay. At this point, the low end for a self repair is probably in the $300 range.
I've purchased an used X200 on ebay in June 2009, with 1 year warranty until October 2010. However, it died on September 2009 (=won't turn on; LED indicators are just flashing up for a couple of seconds and then fade away). Unfortunately, it was model type 7454, and this was the first time i've discovered that international lenovo warranty doesn't mean worldwide (after that case i'm always doing the check, and so far i haven't seen another model type which is not being serviced in Russia).
For a first time, i've thought that this is mistake (all other model types sold in US have their warranty valid in Russia, just not 7454), and came to IBM service, pretending i don't know anything about model numbers, limited international warranty and such. They took off the laptop for diagnostics, and then told me that: they cannot fix 7454 for free because IBM/Lenovo HQ won't provide required parts for 7454; the problem is that motherboard is dead, and needs replacement, and out-of-warranty replacement costs $1800 (at the time, brand new X200 could be easily locally purchased for $1400); and i must pay $70 for out-of-warranty diagnostics. Oh, and, by the way, they "diagnosed" it for two weeks.
Then, i came to another third-party repair service, which also fixes laptop parts (instead of only replacing these with brand-new FRUs, as official lenovo services do), and they diagnosed laptop for free, saying that the root of the problem is that, due of overheat, south bridge "unsoldered" from the motherboard (that is, problem is caused by manufacturer fault). They tried to fix the problem with their methods (except for completely resoldering that BGA chip), but all attempts failed. That took a single day (i've gave them laptop on morning, and collected it back on the same evening), and i didn't pay a penny (and, in case of successful repair, it would cost $100 - compare that to official service demands!)
Of course, after that third-party repair service, motherboard was damaged enough for laptop to ultimately became out-of-service due to unauthorized access and modification of internals. However, i've contacted a friend of mine who lives in USA, and send him the laptop for him to send it as in-warranty item to Lenovo USA. Surprisingly, lenovo didn't asked a question, and in a couple of weeks, the motherboard was replaced by new one, completely free.
PS: Looking at the pictures provided by OP, i fear that there actually might happen a "Liquid ingress". Even if it is foolishly to assume Lenovo had deliberately put some liquid on it to avoid expensive in-warranty repair, it could easily be unintentionally done by some not-so-bright serviceman. I fear that now that laptop may have other problems, not only audio missing (e.g. it may not turn on at all).
PPS: Oh yes, and another IBM service experience. We had several R61i purchased in russia, and one of these died with the very same symptoms as my X200 (won't turn on, although led indicators are flashing up for a while). I've took it to the same IBM service, they said that motherboard died, and replaced it in a couple of weeks.