Doing the math, at least a little
Maximum change in asteroid velocity would require a direct hit normal to the surface.
Deviations from a normal impact will 'waste' energy changing the angular momentum, but not affect the velocity of the body.
Assuming a direct impact normal to the surface (ie... in line with the centre of mass)...
Mass of NASA object ... I believe I read somewhere that it was one ton? one tonne?
Let's go with tonne...
Velocity of DART approximately 6,000 m/sec.
Stony asteroids range from about 3 - 5 gm/cm^3, or 3 - 5 tonnes/m^3.
Lets go with 3 tonnes / m^3.
Diameter of smaller asteroid is about 160 m. Estimate mass at 3 x (4/3) x 80^3 x pi, or roughly 6.5 million tonnes.
Maximum velocity change is thus on the order of 6,000 m/sec / 6.5 x 10^6 or about .00093 m/sec, or .9 mm/sec.
But... this is unlikely to be enough to break up the binary asteroid pair, so gravity will couple the two masses, the second of which is about 780 m in diameter, thus with a mass on the rough order of (780/160)^3 = 116 times greater, implying a probable maximum change in velocity of the asteroid pair of about .9 / 116 = .0078 mm/sec.... or less if the energy gets converted into spin, or potential energy in the binary pair?? (need to think about that some more... it's been a long time since I've had to do physics).
Someone with a better grasp of orbital mechanics will have to tell us how much that will change the orbit of the asteroids...
Yes, I know there are various assumptions of geometry and material, etc, here, but this is order of magnitude stuff, not an exact solution.