I expect details of the mirror are not the point of the question. We will assume a perfectly reflecting mirror, and curve it so all the light is more or less spread evenly over the surface of the Earth.
No this would not be equivalent to a second Sun. As Anna says, the mirror would need to be bigger.

Edit: I have updated the sketch to show an annular mirror. To light up the Earth like a second Sun, it would have to intercept about as much light as the Earth and reflect it evenly on the night side. That means if the mirror was a flat disk, it would have to have about the same surface area as a flat disk the size of the Earth.
Since it is a few thousand miles farther from Earth, the Sun would be slightly dimmer. The area would have to be proportionally larger. Given sunlight intensity follows an inverse square law, the ratio would be
$d_{Earth}^2/(d_{Earth} - \Delta d)^2$.
Given $d_{Earth} = 93,000,000$ miles, this is pretty close to $1.00$.
But both the Earth and the mirror would be curved. The area of the mirror would depend on how the mirror is curved and angled. Without those details, all we can really say is that it would be about the size of the Earth.
If this was a difuse reflector, there would be another inverse square law loss that Anna has covered.
I am assuming a mirror surface. It would reflect all the light on the Earth, and look like an annular sun.