I think it is wonderful that you want to help out your daughter and watch the grandogs. However, there are so many variables with this that I can only ask you questions. You say the surgery will be in the next several months - to me that means March, April or May sometime. If the surgery was at the end of May that would give you two months to begin healing. In that two months you will mainly rest, elevate your leg and ice. You will also most likely be using a walker for short walks around the house then back to icing and elevating. My point being you won't have a whole lot of energy.
Are these big dogs? I wouldn't think you would be able to walk them on a leash, no matter how good of dogs they are. Big dogs that jump up will be a huge problem - your balance will still be not steady. You'll definitely want your walker with you, simply to protect your leg. Little dogs can be a trip hazard also. You'll have to be able to fill water bowls and put them on the ground, and their food bowls. It can be done, I'm just trying to help you think through what all you will be doing. Will you go outside when the dogs go out? If so, are there steps? Sitting outside for a bit several times a day might feel good if there is a comfy chair for you. Hopefully you won't have to police the yard of poo, as that could get very old and difficult quickly.
I think you need to discuss this with your surgeon and also with your daughter. By the way, will she be helping you post-op? She might not realize how involved a TKR truly is, and that recovery is a year long process. Both you & your daughter have to be on the same page about your recovery and your ability to keep the grandogs.