Step 17: Option 2
This switch changes between charging the battery from the solar cells and transferring the charge to the DS.
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150 mAh = .15Ah
.15 Ah @ 9 v = (.15 x 9) = 1.35 watt hours
1200mAh = 1.2 Ah
1.2 Ah @ 3.7 v = (1.2 x 3.7) = 4.44 watt hours
So the answer is no. I'm guessing you are going to use a 5v regulator and putting the 5v through the charging port. Assuming you use a 7805 voltage regulator for 5 volts, which gets about 60% efficiency on average:
1.35 x .60 = about .8 watt hours
If this is correct, it would take the power of about 6 batteries to fully charge it, assuming the charging circuit in the DS is extremely efficient (and it probably isn't) and barely any power is lost through the battery itself. My recommendation: If you are using the 7805 idea, wire six (or slightly more) AA rechargeable batteries together to make a replacement 9v. It will have between 2000 and 2500 mAh (depending on the batteries you use), and should be able to fully charge the battery.
I have found a more efficient way to charge the ds battery.
Diagram
Hope this helps