We present a search for gravitational waves from 116 known millisecond and young pulsars using data from the fifth science run of the LIGO detectors. For this search, ephemerides overlapping the run period were obtained for all pulsars using radio and X-ray observations. We demonstrate an updated search method that allows for small uncertainties in the pulsar phase parameters to be included in the search. We report no signal detection from any of the targets and therefore interpret our results as upper limits on the gravitational wave signal strength. The most interesting limits are those for young pulsars. We present updated limits on gravitational radiation from the Crab pulsar, where the measured limit is now a factor of 7 below the spin...