If you know that k has a prefix or certain fixed bits, the complete value of k can be represented as:

(k= P+X), where X is the variable part (unknown bits). However, if these fixed bits are small, any Lattice attack will still remain unfeasible. This is why I mentioned that you need to compare all possible nonces from the signatures that can be recovered, with the hope of finding a clue, a pattern, a sequence related to something like Mersenne Twister, or differences between the r values.