By some miracle, General Nogi's Third IJ Army is able to breech Port Arthur's defenses and marches up North in time for the battle of Liaoyang. IJA decisively engages and destroys Russian forces there. What would Russia have done next?
The answer to that would depend heavily on the political fallout of that disaster.
You can't really just go look at their OTL response and add to Japanese strengh a bit and reduce Russian strength, then calculate the new results. Well you could, but that woul be a best-case for Russia.