Place the staining solution in a disposable plastic or hazardous waste container under the hood. In 1–2 days the silver metal will precipitate from the solution. Once the silver has precipitated the aqueous waste can then be decanted from the silver and disposed of separately. Disposal of both the solid and aqueous waste should be according to local and state regulations
Most research universities have a system for hazardous waste pickup service where you put the waste in a container marked with a hazardous waste sticker describing the hazards and they take it and figure out how to dispose of it so I would either ask around or search your institution's website for info on that. If it's a kit the SDS will tell you what specific hazardous chemicals are present (usually with silver stain there's either formaldehyde or gluteraldehyde in addition to silver as "stuff that can't go down the drain").
Some university safety departments may provide containers for liquid waste that they will pick up and process. You may have to call them to obtain the container and arrange for a pick-up. Of course the key is accurate labeling and paper work on the stickers/labels and documents that the safety department may provide to you; to accompany the container. Please refer to their guidelines for allowable mixing of the contents that may go into individual containers. They may also have methods to dispose of any solid waste and animal waste.