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Ski Cooper is the shortest drive from Leadville. I have never skied there so I can't tell you anything about it.
Other than that, Copper is the closest resort to Leadville. If I had to guess, I'd say it's about 20 minutes away. You're also about a 30 minute drive from Vail and Beaver Creek.
My favorite mountain is Vail because there are no lift lines if you just want to lap the park, and it has the back bowls + blue sky which are great on powder days.
Although, Breck geographically is right next to Copper, there is no road between the two so Breck is an extra 25 minute drive after you have already passed Copper.
Copper 4-pack is $140 right now, which I would consider a pretty good deal. http://www.coppercolorado.com/winter/plan_and_buy/season_passes
4 day ticket at Vail/Beaver Creek is $360, which isn't such a good deal.
4 day ticket at Keystone/A-basin is $332, also not a good deal.
I think Vail is the best mountain, but the extra money you will be spending isn't worth it. I would recommend going to Copper.
Obviously there is a road between them. What I meant was that there is no direct (relatively straight) route from Copper to Breck. To get to Breck from Copper you have to take I-70 to Frisco, then you have to take highway 9 into Breck.
I just went to google maps to get the driving time between each. I found,
Leadville to Copper: 43 minutes
Leadville to Vail: 54 minutes
Copper to Breck: 25 minutes
Copper to Keystone: 24 minutes