I am glad you were able to get it to work. Your regression is over 0 to 31 while your DateListPlot is over a month in 2007. I am surprised you were able to get that to work.
To put both over days you might try this.
In[1]:= data22 = {
{{2007, 1, 16}, 12582.59`}, {{2007, 1, 17}, 12577.15`}, {{2007, 1, 18}, 12567.93`}, {{2007, 1, 19}, 12565.53`},
{{2007, 1, 22}, 12477.17`}, {{2007, 1, 23}, 12533.8`}, {{2007, 1, 24}, 12621.77`}, {{2007, 1, 25}, 12502.56`},
{{2007, 1, 26}, 12487.02`}, {{2007, 1, 29}, 12490.78`}, {{2007, 1, 30}, 12523.31`}, {{2007, 1, 31}, 12621.69`},
{{2007, 2, 1}, 12673.68`}, {{2007, 2, 2}, 12653.49`}, {{2007, 2, 5}, 12661.74`}, {{2007, 2, 6}, 12666.31`},
{{2007, 2, 7}, 12666.87`}, {{2007, 2, 8}, 12637.63`}, {{2007, 2, 9}, 12580.83`}, {{2007, 2, 12}, 12552.55`},
{{2007, 2, 13}, 12654.85`}, {{2007, 2, 14}, 12741.86`}, {{2007, 2, 15}, 12765.01`}, {{2007, 2, 16}, 12767.57`}};
dat0 = AbsoluteTime[data22[[1, 1]]];
dat1 = AbsoluteTime[{1900, 1, 2}];
data33 = Map[{(AbsoluteTime[First[#]] - dat0)/dat1, Last[#]} &, data22]
Out[4]= {
{0, 12582.6}, {1, 12577.2}, {2, 12567.9}, {3, 12565.5},
{6, 12477.2}, {7, 12533.8}, {8, 12621.8}, {9, 12502.6},
{10, 12487.}, {13, 12490.8}, {14, 12523.3}, {15, 12621.7},
{16, 12673.7}, {17, 12653.5}, {20, 12661.7}, {21, 12666.3},
{22, 12666.9}, {23, 12637.6}, {24, 12580.8}, {27, 12552.6},
{28, 12654.9}, {29, 12741.9}, {30, 12765.}, {31, 12767.6}}
In[5]:= nlm = Normal[NonlinearModelFit[data33, a + b*(t)^0.3, {a, b}, t]]
Out[5]= 12479. + 60.8924 t^0.3
In[6]:= Show[ListPlot[data33], Plot[nlm, {t, data33[[1, 1]], data33[[-1, 1]]}]]
Out[6]= ...PlotSnipped...
That at least has both the points and the regression on the same axes.