The best (IMHO) Italian food in Santiago is in Providencia, at Rivoli. It's found along a pedestrian walkway that connects Nuevo Lyon (the northern extension of Av. Ricardo Lyon) with the rowdy nightclub district of Suecia. It can be tricky to find, so ask around. It's worth it.
There is also an outlet of Astrid y Gaston in Providencia, I forget the street but everyone at your hotel will know it.
I also recommend you try El Liguria, a bar that serves good food. There is one outlet in Providencia and one in Las Condes around the corner from the Tobalaba metro station.
Near to Providencia you can find good food in the El Bosque Norte area of Las Condes (Argentine steakhouse Happening, La Nolita on Goyenechea and Osaka at the upscale W Hotel come to mind). This area is about 20 minutes walk from your hotel, or a couple of metro stops (get off at El Golf and exit to the north side of Av. Apoquindo).
If you want to venture a little farther there are a bunch of excellent restaurants in Vitacura, which is northeast of Providencia, a bit far to walk but a short cab ride. La Mar on Av. Nueva Costanera is excellent for Peruvian-style seafood (which is much better than the Chilean style, in general), and Europeo on Alonso de Córdoba is one of the finest restaurants in the city (and priced accordingly).
Good food has not traditionally been one of Chile's strengths but Santiago has come a long way in recent years and there are now many good restaurants, with new ones opening all the time. A bonus is that most have an excellent selection of good Chilean wines, and the markup on wine is much less than what one expects state side.