I take it you're referring to Nicole, Admiral Paris's assistant (or secretary), you honestly think a Starfleet Admiral isn't going to have a staff? Nicole likely managed the Admiral appointments and kept him on schedule.
Which is utterly idiotic given that a computer (a Trek one no less) can do that orders of magnitude better.
Roddenberry was the one who gave both Captain Pike and Kirk a yeoman, a yeoman is basically a secretary.
But you were saying.
A yeoman still served on board a star-ship and had numerous other duties who probably had a temporary assignment as a yeoman.
Besides, everyone who serves under a captain is a personal 'secretary' of sorts, or people who get things done for them.
On a star-ship, you can get away with it partly (even though its unnecessary and contradicts numerous things)... for a planet bound setting, especially in the late 24th century... its just downright stupid (we can do away with such stupidities TODAY with our automation capabilities, let alone Trek).
We don't know how long it would take for the Doctor to reach Federation space. Assuming this planet is not a member of the Federation, he might have had to do a bit of searching for the nearest Federation world. Still, that might not have been too far away.
It was mentioned that the Doctor attempted to trace Voyager's path.
However that still doesn't negate the premise that he might be able to stumble upon a star-fleet vessel or an outpost.
Besides, warp travel would likely advance to the point where it wouldn't be an issue even for the Doctor to reach Earth in less than a day or so (but it wouldn't surprise me that this particular planets technology was in stagnation - seeing how tensions were 'on the rise' for 700 years - and for a technologically advanced species no less).