Here is the code cleaner. Sorry about that:
ClearAll
SetAttributes[{vhat, khat, w, c, xin, \[Delta], v, j}, Constant]
khat = .162
xin = .028
vhat = .017
Clear[yOUT, yIN, ytest, nIN, nOUT, WronsFUN, \[Lambda], w]
WronsFUN[\[Lambda]_, w_, xoo_] := Module[{val}, Clear[nIN, nOUT];
nIN = NDSolve[{\[Lambda] == -I*w*x^(3/2) - vhat*x/y[x]*y'[x] -
7/2*khat*x^2/y[x]*y'[x] - khat*x^3/y[x]*y''[x],
vhat*xin^(-1/2)*y[xin] + khat*xin^(3/2)*y'[xin] == y[xin],
y[xin] == 1}, y[x], {x, xin, 1}];
yIN[x_] = First[y[x] /. nIN];
nOUT = NDSolve[{\[Lambda] == -I*w*x^(3/2) - vhat*x/y[x]*y'[x] -
7/2*khat*x^2/y[x]*y'[x] - khat*x^3/y[x]*y''[x],
vhat*y[1] + khat*y'[1] == -y[1], y[1] == 1}, y[x], {x, xin, 1}];
yOUT[x_] = First[y[x] /. nOUT];
val = yIN[xoo]*yOUT'[xoo] - yOUT[xoo]*yIN'[xoo]; val]
FindRoot[WronsFUN[j, 21.67, .038], {j, .1 - .1*I}]
WronsFUN[7.07812 - 1.98627*I, 21.67, .038]
