Attack with longsword: 1d8 (longsword) + 1d8 (improved smite) + STR, x2 dice for critical because target is paralyzed in melee range = 4d8 + STR
Rider/smite (4th level slot, cost: 6 SP): 5d8 (base) + 1d8 (improved smite), x2 dice for critical = 12d8
Total damage on hit: 16d8 + STR
Extra attack means he does the longsword twice, so 32d8 + Str x2 = 32 * 4.5 = 144 + Str x 2, with Str 16 is 150 damage. You could swap out your longsword for a greatsword to boost your damage slightly but it hardly seems necessary.
You added Improved Smite in twice for each attack or four times total. ((1D8 weapon + 5D8 smite + 1D8 improved smite) * 2 + Str) * 2 = 28D8 + 4x Str = 132 damage with 16 Str.