Take very long time to solve the nonlinear 1-D PDE with NDSolve

I am trying to solve the nonlinear PDE, recently. The formulas are given as

with the conditions

I write the numerical code to solve it. The code is written as

sys = {b*(u[r, t] - m[r, t]) + 
     D[u[r, t], {t}] == (-D[u[r, t], {r}])^((1 - n)/n)*
           ((kd^(1/n)*D[u[r, t], {r}, {r}])/
        n + (kd^(1/n)*D[u[r, t], {r}])/r), 
       l*D[m[r, t], {t}] == b*(u[r, t] - m[r, t]), 
   Derivative[1, 0][u][rw, t] == -(2^n/(kd*rw^n)), u[R, t] == e, 
       u[r, e] == e, m[r, e] == e}; 
sol = NDSolve[sys, {u, m}, {r, rw, R}, {t, e, T}, 
  MaxStepSize -> {0.01, 1}, 
     Method -> {"MethodOfLines", 
    "DifferentiateBoundaryConditions" -> {True, "ScaleFactor" -> T}}]

To verify the numerical results, the semianalytical solution is used. When n =1, the equations are linear. The code is

Vi[n_, i_] := 
 Vi[n, i] = (-1)^(i + n/2) Sum[ 
     k^(n/2) (2 k)! /( (n/2 - k)! k! (k - 1)! (i - k)! (2 k - 
            i)! ), { k, Floor[ (i + 1)/2 ], Min[ i, n/2] } ] // N; 
Stehfest[F_, s_, t_, n_: 16] :=
 If[n > 16, Message[Stehfest::optimalterms, n];
        If[ OddQ[n], Message[Stehfest::odd, n];
                "Enter an even number of terms",
                If[n > 32, Message[Stehfest::terms, n];
                    " Try a smaller value for n. Maximum allowable n is 32 ",
                    Log[2]/t Sum[ Vi[n, i]*F /. s -> i Log[2]/t , {i, 1, n} ] ]],
        If[ OddQ[n], Message[Stehfest::odd, n];
                "Enter an even number of terms", 
    If[n > 32, Message[Stehfest::terms, n];
                    " Try a smaller value for n. Maximum allowable n is 32.",
                    Log[2]/t Sum[ 
       Vi[n, i]*F /. s -> i Log[2]/t , {i, 1, n} ] ]]]  // N; 
s[r_, t_] := 
 Stehfest[(2^(2 + n) r^(1/2 - n/2)
      rw BesselK[(-1 + n)/(-3 + n), -((
       r^(3/2 - n/2) Sqrt[(2^(1 + n) n p (b + p l + b l))/(
        kd (b + p l))])/(-3 + n))])/(kd p (rw^2 Sqrt[(
        2^(1 + n) n p (b + p l + b l))/(kd (b + p l))]
         BesselK[
         2/(-3 + n), -((
          rw^(3/2 - n/2) Sqrt[(2^(1 + n) n p (b + p l + b l))/(
           kd (b + p l))])/(-3 + n))] + 
       rw^2 Sqrt[(2^(1 + n) n p (b + p l + b l))/(kd (b + p l))]
         BesselK[(
         2 (-2 + n))/(-3 + n), -((
          rw^(3/2 - n/2) Sqrt[(2^(1 + n) n p (b + p l + b l))/(
           kd (b + p l))])/(-3 + n))] + 
       2 (-1 + n) rw^((1 + n)/2)
         BesselK[(-1 + n)/(-3 + n), -((
          rw^(3/2 - n/2) Sqrt[(2^(1 + n) n p (b + p l + b l))/(
           kd (b + p l))])/(-3 + n))])), p, t, 16]

I compared the results and found that the curves predicted by numerical solution and semianalytical solution are close to each other. It represents the MaxStepSize is valid.

The parameters is given as

kd = 10; 
n = 1; 
T = 1000; 
rw = 0.025; 
R = rw + Sqrt[(Pi*T)/1.4]; 
b = 0.5; 
l = 50; 
e = 5./10^10; 

With the result

S0 = {#, s[5/10, #]} & /@ Exp@N@Range[-2, Round@Log@T, 1/4];
ap1 = ListLogLogPlot[S0]; np1 = 
 LogLogPlot[u[5/10, t] /. sol, {t, 0.1, T}]; Show[ap1, np1, 
 PlotRange -> All]

However, as n is larger than 1 (e.g., n = 1.1, 1.5, and 2), the procedure of the solving the nonlinear equation is stuck. Is there anything wrong in my numerical code? How can I fix it? Thank you.