The simple answer is: You reduce a cream sauce the same way you reduce any other sauce, by simmering it until a certain amount of liquid is gone, just like the instructions said. You have to be careful about temperature though, because milk (or cream) can burn at high temperatures, and then your sauce is ruined. You should keep it to a low or at most medium simmer.
Cream sauces normally tend to thicken extremely fast, so the long cooking time is almost certainly due to watering it down. I don't think the water was necessary at all; the chicken broth was probably for flavouring, and although "broth" is a somewhat nebulous term, one would normally expect a broth to contain at least some amount of gelatin, which will cause the sauce to thicken substantially when it cools if you reduce it a lot. Water doesn't, so you've added no flavour and thinned out the sauce.
Basically, you (or your wife) added water for no other purpose than to try to evaporate it later. Water generally doesn't go in a cream sauce. If you don't have chicken broth or can't use it, I would either substitute more wine or just leave it out completely. Usually the only time you substitute water for broth is if it's actually the base of your sauce.
As the others have mentioned, it won't quite be the same ... however, it can be done, it just won't quite have the same flavor as there isn't a chance for the meat juices to transfer into the sauce, and some issues regarding mixing it back in with the cold pasta.
If I were going to try it, I'd brown the meat, remove some of the fat if it's a really fatty grind, then add some extra sauce (or even other liquid ... maybe dairy, for a classic bolognese) to let it simmer for a bit ... then toss the pasta in with it to reheat.
The extra sauce is the key, otherwise, the meat isn't going to blend in with sauce on the pasta, If you reheat the pasta in the fresh sauce, it should hopefully mix together more completely.
... but if I were to do it, and wanted to add a meat to an already cooked pasta, I'd probably not choose ground beef as a first choice; I'd probably cook up some sausages, cut it down to an appropriate size for the pasta you're dealing with, and mix that in, possibly adding some extra liquid when reheating the pasta.
Best Answer
First, the alcohol doesn't burn off. We had a table about the percentage of alcohol left after a period of cooking, and especially in something cooked as short as a pasta sauce, there is a substantial amount left. For the longer discussion, see Cooking away alcohol.
Second, alcohol is a great solvent. It can leach aromatics from spices and herbs which wouldn't come out with water only. There is a good reason why extracts and essences are made with alcohol. While a really good extract needs a long time, even the short cooking time will be enough for alcohol to make a difference.
Third, it can stabilize your sauce. Cream is an emulsion of fat in water, and not terribly stable, especially if some acid is present. Alcohol dissolves both fat and water, so it will stop the cream from eventual separating.
Fourth, maybe you happened to find a not so great recipe. Vodka will help with the first three points, but it is seldom a good choice for cooking, because there is almost always another alcohol which will accomplish the same and introduce a good flavor which harmonizes with the remaining ingredients. For pasta sauces this tends to be a good brandy, but it depends on what else you have in the sauce. And because even vodka will leave a taste of alcohol in the end, it is hard to argue that it can be used where the taste should be kept unchanged.