Short answers, pulled from the story.
The Battle of La Rothière took place on the 1st of February 1814. Napoleon reached the theater of action on the 26th of January and assumed command of 41,303 available soldiers before the engagement began.
Napoleon commanded approximately 45,100 men with 128 guns while the Coalition engaged about 85,000 troops and 200 guns in the battle. Historian George Nafziger stated that 78,000 Allies fought out of 118,000 men available against 41,000 French forces.
Field Marshal Karl Philipp, Prince of Schwarzenberg led the Army of Bohemia and Gebhard Leberecht von Blücher commanded the Army of Silesia during the conflict. General Ferdinand von Wintzingerode moved his North Army across the lower Rhine on the 6th of January to support the invasion.
Historian Digby Smith stated that French losses numbered 4,600 killed and wounded plus an additional 1,000 soldiers and 73 guns captured. The Coalition had between 6,000 and 7,000 casualties according to Smith while other sources like Chandler assigned losses of 6,000 casualties to each side.
The weather turned terrible during the engagement causing heavy ground that made Napoleon's artillery nearly useless. Roads were in such poor shape that Sacken doubled teamed half his 72 artillery pieces and left the rest in the rear while snowfall disrupted French formations.