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Since $n$ and $k$ are both bounded, you can try something like this:
sage: var('a b c d n k')
sage: nmin = ceil(1080/90); nmax = floor(2000/60)
sage: kmin = ceil(1000/70); kmax = floor(1920/40)
sage: constraints = [a >= 80, b >= 1000, c >= 20, d >= 40, a + b <= 2000, c + d <= 90]
sage: result = []
sage: for nn in [nmin..nmax]:
....: for kk in [kmin..kmax]:
....: result.append( solve(constraints + [n==nn, k==kk, a+b==n*(c+d), b==k*d], (a,b,c,d,n,k)))
I think this will take some time to run, but it should get you all the solutions you want.
