The values of k for which this has been done are those at least 1014. In some concluding remarks Bruce Reed says: "We note that fairly straightforward modifications yield the value 108. We are certain that the proof can be modified to yield 106. We expect that a more careful analysis could bring the bound down to 1000. However, we doubt if the proof technique can be used to prove any bound better than 100."
In the paper Bruce Reed shows that the following statement 23 implies the full Borodin-Kostochka conjecture, and he remarks that this would be a very attractive way of proving the conjecture. (However he does not conjecture that 23 is true).
23 Every graph G of maximum degree nine all of whose cliques of size eight are disjoint, can be partitioned into two graphs of maximum degree four, neither containing a clique of size five (and hence by Brooks' Theorem such a G is eight colourable).
Bjarne Toft, September 30, 1997.