Determining which metal is strongest is difficult to answer because the criteria for "strength" will depend on context. Generally, materials scientists refer to the tensile strength (the force required to break a material), which can be broken into yield strength (the force that can be applied reversibly, without deforming the material) and ultimate strength (the stress required to break the material apart). There are standard testing conditions used to generate these values, since materials can behave differently based on how the stress is applied. Not all materials are brittle (breaking under low stress). Indeed, many metals will stretch/deform in response to stress long before breaking (having low yield but high ultimate strength).

Based on your criteria (density and impact resistance), tungsten is the strongest pure metal. There is a correlation between density and strength but there is significant variation between materials - check out this chart:

strength-density

The strength of tungsten also comes from the geometric arrangement of its atoms (its crystalline lattice), not just its density or the strength of the bonds between particular atoms. However strong, tungsten does not have a high strength-to-weight ratio - this is why you see titanium being used more often when high strength is needed. This is a good discussion of how multiple metals can be considered "strongest":

strongest metal-hardest metal

Notably, mechanical failure is usually the result of impurities and defects in the lattice structure which lower the tensile strength locally, even if the material is theoretically strong. Alloys, which are particular mixes of metals, can be stronger than their component materials, in part because they can better eliminate defects and voids in their lattices.