They've published results using 1 to 1,7/fb of data by now. Discovering the most difficult Standard Model Higgs with a mass of 115 GeV would take an average of 17/fb, although there would be hints long before. By now they've gathered 2,5/fb for both CMS and Atlas. By the end of this year's running (end of October) they will have gathered 5-7/fb each. These can be combined for a total of 10-14/fb which should be enough to give us a strong hint of whether the Standard Model Higgs is there or not.
LEP excluded everything up to 114 GeV and the Tevatron excluded everything above 158 Gev. This allowed range of 114 - 158 GeV has now shrunk to 114 - 145 Gev so about 30% of the allowed range has been cut by the LHC. Add to this that indirect measurements show us the Higgs is more likely to be on the light end of this interval than the heavy end.
Also note that all this is specific to the Standard Model Higgs while what you call an "almost foregone conclusion" would be (I think) that there is some Higgs-like mechanism to be found. The mathematics require that something fixes the Standard Model at these energies: the probabilities no longer add up to one.
The most popular alternatives, different flavors of supersymmetry, have several Higgses, usually with a lower cross section than the Standard Model one. That means you need more data to find them. IIRC, the worst case for the simplest supersymmetric model, MSSM (Minimal Supersymmetric Standard Model) is that you need 300/fb for a discovery. Although that's more than we'll get by the end of 2012 (after which there's a break for the upgrade to 14 TeV) it still means we would have a small 2-3 sigma bump by then.
If there's no Standard Model Higgs we'll know by the end of this year. Many, if not most, physicists would call this the most likely outcome.