After a pretty close race, the 
1911 New York Giants fought off the 1912, 1913, 1904, and 1905 editions of the 'Jints to win their qualification league and represent the New York Giants franchise.
The final standings at the top:
  W  L  Pct.  GB
1911  89  60  .597  -
1913 88  64  .579  2.5
1904 87  65  .572  3.5
1912 87  65  .572  3.5
Despite winning the World Series in a 4-game sweep, the 1954 Giants finished in 20th (that'd be last), 28.5 games back.
The hitting categories were mostly led by teams from the 1920s and 1930s, as one might expect, save for 
Jack Doyle of the 1894 squad sitting atop the batting average and obp categories and  
just missing a .400 season (.399).  
Mel Ott (1934) led in slugging and runs created, Rogers Hornsby (1927) in runs scored, George Kelly (1922) in RBI, and 
Bobby Thomson (1951) hit 33 shots heard 'round Kings Contrivance to edge out Ott (1936) and lead in home runs.
Pitching statistics were biased by the 1894 team, who had two pitchers starting 70%+ of their games in real life.  Those two pitchers (
Amos Rusie and Jouett Meekin) both topped 30 wins.  Hall of Famer 
Christy Mathewson took up 4 slots in the top 10 winners 
and top 10 ERA, with the 1905 edition performing best: 29-10 (3rd in wins, 2nd in win pct., 1st in ERA, 4th in strikeouts).  
The 1911 team didn't have many statistical leaders, but managed to grind out the pennant by being among the team leaders in batting and pitching.  In reality they lost the World Series to the Philadelphia Athletics.  
The Orioles league has stalled out a bit (naturally, since I'm actually trying to manage one of the teams instead of letting the computer do all the work).  Not sure if the next all-computer league will be Oakland or Minnesota...