There seems to be quite a bit wrong with that.

1) There was no differential equation for E2 so I made one up.

2) There was no value given for kin.

3) The derivative signs were incorrect.

4) It would be good style not to capitalise Model.

5) Also, you use numerical integration so I gave all variables a finite precision.

This will work to some extent:

ClearAll["Global`*"];

k1 = 600.;
k2 = 6000.;
k3 = 10.;
k4 = 0.5;
k5 = 100.;
k6 = 100.;
ki = 10.;
Et = 10.;
S0 = 50.;
kin = 20.;


Model = NDSolve[{Eu'[t] == k2*ES1[t] - k1*Eu[t]*S[t] + k3*ES1[t] + k4*ES1[t] - k6*Eu[t]*S[t] + k5*ES2[t], 
   ES1'[t] == k1*Eu[t]*S[t] - ES1[t]*(k2 + k3 + k4), 
   ES2'[t] == k6*Eu[t]*S[t] - k5*ES2[t] - kin*ES2[t], 
   P1'[t] == k3*ES1[t], P2'[t] == k4*ES1[t], 
   S'[t] == k2*ES1[t] + k5*ES2[t] - k1*Eu[t]*S[t] - k6*Eu[t]*S[t], 
   E2'[t] == -k4, ES2[0] == 0., S[0] == S0, Eu[0] == Et, ES1[0] == 0., 
   P1[0] == 0., P2[0] == 0., E2[0] == 0.}, {Eu, ES1, ES2, P1, P2, S, E2}, 
   {t, 0, 60}, MaxSteps -> 1000000, PrecisionGoal -> \[Infinity]]

You will need to correct the equation that I made up for E2 and fix the parameters. The results can be plotted but look very nasty right now as expected.

Cheers, M.