How can I get all the water to go down the drain in this sink?
Hard to tell from that picture, but you may be able to shim the cabinets to make the sink and countertop more level.

If not, you can (if the sink bottom is not too thin) grind it (diamond will take on anything, but silicon carbide is a lot cheaper, so try that first) to fix the slope - but it looks like you'll also have to remove and reseat the drain - the strainer appears to sit up from the sink bottom, where it should sit down into it for proper drainage.