Kornilov would most certainly push Russia to stay in the war as long as possible. With the example of the Kerensky Offensive, however, by this point the Russian Army was ineffective. If Kornilov wants them on the offensive, they'll most likely falter. If he just wants to keep a defensive line, which is smart of him and more likely, well, the Germans will still most likely make the gains they did in OTL, all the way into the Ukraine, up to Estonia, etc. etc. That Russia will stay in the war up to the very end makes me think that the Germans have a good chance of taking Petrograd, landing troops in Finland, cutting off supply routes in North Russia, and connecting with the Ottomans in the Caucasus. At that point, they'll most likely have reached the end of their logistical limits and engage stiff resistance from the Moscow region. No doubt the Spring Offensive would never happen, which means an earlier victory for the Allies could happen. However, can they force Germany to surrender in time to include Russia in the peace deal, especially when Germany has so much territory back East to lord over? Germany might just decide to buckle down, replacing factory workers with POWs and sending more German troops to the front.
But lets go back. After taking power, Kornilov would probably begin by throwing everyone associated with the Petrograd Soviet into prison, including perhaps even Kerensky. He would then take over the Provisional Government, likely reorganize it with some of his loyalists, but the bureaucracy would stay the same, as would most of the main figures in the PG, they would just be marginalized somewhat. Kornilov wasn't a monarchist... he was a republican. He would have fought to reintroduce the duma and perhaps a president as chief and head of state to replace the Tsar. The Republic of Russia. He would also push for as many territorial rewards as possible if he could hang on until victory... Constantinople, the Straits, Armenia, Galicia, etc.
But what of the Bolsheviks? If things get bad as I think they will, with the Germans occupying Petrograd and pushing on Moscow, I think they'll be given even more support. They might still succeed in a putsch, now based in Moscow, against a weak and unpopular Kornilov government, to take Russia out of the war. Certainly Kornilov will have to put down an attempted coup at least once, more likely twice, in order to include Russia in the Peace Conference.