So he should be free from the nuke threat for some time now. Good. Keep it that way!!! otherwise they are going to blow a hole into the Korean front and annihilate the US army yet again.
I would now try to drive them out of Australia, and then turn the tide in the air war. Maybe allocate one nuke to Australia, and two to operations needed for the air war... like, one for the reconquest of Philippines (try to bag some Soviet air squadrons there!!! Paras are your friend!) and then one for the liberation of Taiwan.
After that focus all production on holding the line in Korea, and winning the air war. Stack aircraft onto the regained bases in the Philippines, Japan and Taiwan. Favor naval bombers over ships, naval bombers are more versatile and you do not lose such a huge investment in one swoop if you suffer a defeat.
Stockpile the nukes while you can, and once the hordes of aircraft are pushed back, try to break out from Korea. Then use the nukes. Use them when appropriate... one or two for softening up the enemy line, then all the rest for destroying pockets of resistance.
This is how the Soviets planned to win WW3 in Europe: Break through NATO's forward defense using conventional and limited atomic means, then put all effort into keeping the war fluid and mobile, and push NATO into the German cities and built-up areas. Avoid combat in those areas, nuke them to wipe out the pockets, so that the advance does not get bogged down. Overrun west Germany and reach the Rhine seven days after the start of operations. Even if NATO escalates to all-out nuclear war and communications with Moscow are lost, do not stop the advance.
Once you have turned the tide in the air war and established strong NAV bomber patrols over the vital sea lanes, this is how you should approach the Manchurian and Chinese theatres. Defeat the AI with maneuver warfare, using the atomic bombs to augment your forces' hitting power. Avoid getting bogged down because the AI has a lot more units than you, and if you stand still he will grind you to dust. So keep the maneuver fluid, encircle large pockets, and then quickly nuke them. That way you could grind down the Soviet army and get into new positions deep in China, where you could now establish friendly puppets or carry out a sustained bombing campaign of the Soviet production centers. If you manage to free north-central China and put the industrial centers of the USSR under Hanoi-style bombardment, that's when I would call the war won and trigger peace events.
In summary:
1) Drive the Sovs out of Australia by whatever means necessary (1-2 nukes if needed)
2) Crank out aircraft and re-invade the Philippines, Taiwan (2 nukes if necessary)
3) Deny the seas of Japan and China to the Soviet air forces by establishing a strong US air force there (no nukes unless needed to secure a substantial share of the Soviet air force)
4a) Re-open maneuver warfare by breaking out of Korea into Manchuria, Soviet far east provinces and northern China (use all your nukes there)
4b) Establish the line "Amur River - Mongolian border - Huang He river" as operational limits for the breakout from Korea
4c) If possible, invade southern China as well, but only if this does not come at a weakening of the Korean operation: Aim for a line Hong Kong - Nanjing as operational limits on the southern Chinese mainland. The purpose of this line would be to set up forward air bases to secure the Chinese seas, also to deny the industrial centers of southern China to the Soviets and secure them for a possible Chinese puppet government.
5) End the war through negotiations with the USSR. US negotiation to secure the following points for a lasting peace between USSR and Chicago pact in Asia:
- China to be let free from Soviet rule, and a neutral government to be set up at Chonqing.
- The coastal regions and Manchuria to come under a pro-US government. Sinkiang and Mongolia to remain with the Soviets.
- The formerly Russian far east provinces (everything east of the Amur river) to come either under US occupation or under a neutral Russian government. (i.e. demilitarization)
- Southeast Asia (Vietnam, Cambodia, Laos) to be neutralized under national governments.
- India to remain under Soviet rule.
This should secure peace for the time being, and give your allied governments some breathing room. Japan could see some reconstruction events (raising Infrastructure and adding some IC) and crank out the air force that will defend the pro-US east asian sphere. Your pro-US government in coastal China would produce the infantry to man the fortifications and let you redeploy most of your troops elsewhere.