First, I'd like to remark that there is a computer programs that will tell if a knot is unknotted, namely Jeff Weeks Snappea. It's not rigorously proven to work as is, but in practice it always works.
The main difficulty with Haken's algorithm is the immense size of the objects that must be surveyed. For a 3-manifold described by gluing $n$ simplices together, there are $3^n$ integer programming problems that among them represent every possible incompressible surface in the manifold, each one of dimension linear in $n$ (details depend on exactly how this is implemented). Each of these integer programming problems in itself is challenging: each has a finite semigroup basis for the solution set, but the size of the basis grows exponentially with $n$. The exponent of growth is sufficiently high that it becomes rapidly hopeless. For simple knot diagrams, everything is already known, and there's not much point. If you get up to knot diagrams with perhaps 20 crossing, then there would be perhaps $3^20 > $$3^{20} > $3 billion integer programming problems, each involving very many fundamental solutions with perhaps 100-digit integers that are not easy to identify. There may be various shortcuts and simplifications, but improvements only will increase the complexity of manifolds that can be analyzed by a little bit.
In contrast, Snappea can easily handle knot projections with a hundred or two hundred crossings very quickly, orders of magnitude more quickly than the time it takes to draw a picture of the knot.
