I use a similar system on a product I recently worked on to monitor the battery voltage - and of course I didn't want it on all the time to drain the battery.
The trick, though, is to do the exact opposite of what you are doing. Instead of isolating the ground (which leaves the battery then directly connected to the Arduino) you need to completely isolate the battery from the divider.
This can be accomplished by using a P-channel MOSFET which you can then drive with an N-channel MOSFET:
The theory is, when the Arduino DOUT is either LOW or not connected (pulled low by 10K resistor) the N channel MOSFET is turned off. That means the gate of the P channel MOSFET is being pulled up to the battery voltage by the other 10K resistor, so the P channel MOSFET is off, which isolates the whole divider from the battery.
When DOUT goes HIGH it turns on the N channel MOSFET which then pulls the gate of the P channel MOSFET down to ground which then turns it on, thus connecting the divider to the battery.
When the P channel is OFF the only circuit connected to the ADC is the connection between the ADC input and ground through a 10K resistor, so everything is happy and safe.
You should not use resistors greater than around 10K in size on the input to the ADC because the ADC itself has a resistance (the impedance) which is placed in parallel to the lower resistor of the divider. If the divider's resistors are too large that impedance will affect the accuracy of your readings because the total resistance of the lower resistor will be skewed slightly. Yes, it's possible to compensate in software, but as long as you keep to a limit of 10K or so for the resistors you won't need to worry about doing such things.
Note that it is important to use a P-channel MOSFET for this, not a PNP transistor. A P-channel MOSFET, when on, looks like a very small resistor. A PNP transistor is seen as a fixed voltage drop skewing your results. The N-channel MOSFET can be replaced by an NPN transistor if you wish, it makes no real difference. I nearly always use MOSFETS these days instead of BJTs though as a matter of course.
As an aside, the only other way of achieving this whilst maintaining the isolation of the ADC from the battery would be to replace the P channel MOSFET with a small relay (say a reed relay), but then you still have to build a relay driving circuit which is just about as complex as this one.