Quote Originally Posted by Misha Tal
Синцов заворочался и проснулся с той же мыслью, с какой заснул.
My humble knowledge of Russian grammar tells me that a declension of "который" should be used, instead of a declension of "какой"; that is to say, it should be "... с которой заснул" instead of "...с какой заснул".
This is an archaic phrase. In contemporary Russian "той, которой" is better.

Quote Originally Posted by Misha Tal
"Я знаю того, с кем вы говорили"
As was told, "тот, кто" is applicable in the sense of "the one who".

Another archaic phrase is "с той, что", "с тем, что" can be used in both senses in both sentences above.