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RoboPy
Python module for robotics. Covers transforms, kinematic models of serial link arms created with DH parameters, dynamics algorithms like RNE and Euler Lagrange formulations, and tools for visualizing and simulating robots.
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Check out https://github.com/uw-biorobotics/IKBT
You can get good performance with relatively simple algorithms, usually iterative. There are analytic algorithms, that is algorithms where you have an exact equation for the solution, but those only exist for specific types of robot arms, where you have a spherical wrist as the last 3 links. Iterative methods exist though, which will solve quickly enough for a desktop toy. I see that some others have already posted example code; the MATLAB Robotics Toolbox by Peter corke has ik algorithms prewritten, though you'll have to make a robot model using his code (not too bad, if you have used DH parameters before). I've written a similar library in Python here.