Di you say if you have ever been to Europe before? You are going to visit large, prosperous cities in a wealthy, developed country. I only say that to point out that you are deceiving yourselves if you believe you will be driving the car from museum to restaurant to hotel to viewing platform, parking at a free curb space each time. That's not going to happen, and not only because European cities have, in some cases, managed to escape the US conceit that a car is a like a horse in an 1840's western movie!
Think about what happens when you visit Chicago or Cleveland or Denver. That's more like the driving experience in those Danish cities. Even if you don't care about paying $20 every time you go to a garage, some of them may be valet garages with a big time delay to get the car back each time. You will be surprised how narrow the aisles and parking stalls in most European city garages are. A Chevy Suburban will not fit in most of them, even if you can rent one and drive a standard transmission.
In any European country, there is sometimes good reason for a car, when rural locations, church birth records, and barren wartime sites are the targets. But in the absence of detailed research on where you are going to take your mother, you have to tentatively assume that a car will be a hindrance, not a help.
A big question is whether she can climb the steps to trams and busses. But cities have taxi companies, too. You need to budget the car rental money for the taxis. It is a problem to fit 5 people in most cabs, however. It's worth trying to find out if there are minivan or similar options in your cities of choice.