If you sketch out the triangle of velocities you will find that to obtain a closing speed (or cross track speed) the equation is:
Required ground speed = Required closing speed / Sin of closing angle
You have specified a closing speed of 1 nm per minute which is 60 knots, so the equation becomes:
Required ground speed = 60 / Sin of closing angle.
Signs for most practicable angles are less than 1 so mentally dividing by the sign is a bit difficult. So to simplify the task we can multiply by 1/ the sin.
This changes our equation to:
Required ground speed = 60 x 1/Sin of closing angle.
Taking a few angles as examples
Sin 40 degrees = 0.642787609
1/Sin 40 = approximately 1.556
We now have:
Required ground speed = 60 x 1.556
This is probably a bit too difficult to work out mentally so we need to simplify it. We can do this by rounding up to 1.6 to give us
Required ground speed = 60 / x 1.6 which is 96 knots
So if we are 1 nm off track and we use a 40 degree closing angle we need approximately 93 knots to get back on track in one minute.
To test the accuracy use the original equation
Required ground speed = 60 / Sin of closing angle.
Required ground speed = 60 / 0.642787609 which is 93.34 knots.
If we go through the same process for a few other angles we get:
50 degrees Required ground speed = 60 x 1.3 = 78 knots (78.3).
45 degrees Required ground speed = 60 x 1.4 = 84 knots (84.9).
40 degrees Required ground speed = 60 x 1.6 = 96 knots (93.3).
35 degrees Required ground speed = 60 x 1.7 = 102 knots (104.6).
30 degrees Required ground speed = 60 x 2.0 = 120 knots (120).
25 degrees Required ground speed = 60 x 2.4 = 144 knots (141.9).
20 degrees Required ground speed = 60 x 2.9 = 174 knots (175.4).
All of these solutions clearly include some rounding errors (the more accurate figures using 60 / sin are in brackets. If you wish reduce these errors and avoid having to do lots of mental arithmetic in the air, just select a few angle and calculate the required ground speeds using your calculator and the full equation:
Required ground speed = 60 / Sin of closing angle.
Then write down the required ground speeds for the selected angle on your knee pad (or whatever you use). You can read the required ground speeds directly from your knee pad.