It sounds like the situation (and your daughter) are suffering for lack of consistent, appropriate discipline and an understanding of the cause of your daughter's behavior.
First, get a discipline plan in place. You, your spouse, the nanny, and anyone else involved in your child's upbringing must stick to it or it won't work. It should go something like this:
Warning -- in a firm voice tell her to stop doing X or she will go to time out. (In the case of hitting, the warning comes before she hits, when she shows signs of getting ready to hit.)
Time out -- If the behavior happens after the warning, get on eye level with her, tell her that she is going into time-out because she did X, then pick her up and put her in the time-out spot. She stays there for two minutes (1 minute per year of age) and if she leaves time-out you put her back without a word (this is important -- DO NOT engage her) and restart the timer. At first, this will be a marathon event. Once she's figured out that you won't back down no matter how long she fights it, she'll stay put. Don't give up.
Apology -- When time out is over, get on her eye level again. Ask her why she was in time-out and if she can't or won't explain, remind her firmly. Then have her apologize, to you and to the child she hurt. After that she gets hugs and kisses and life goes on.
It is extremely important that this is done with unfailing consistency, and not just for hitting. Make a list of house rules and make sure they are followed. While less severe behaviors (such as running in the house) will garner a warning the first time the behavior is done, not before, the rest of the process stays the same. The best way to prevent severe misbehavior is to stop it before it gets severe.
The second part of this process is to figure out why your child was behaving this way in the first place. A failure of discipline certainly led to this becoming a regular habit, but it started for a reason. Is your daughter reacting out of jealousy when the other child gets attention? Is the other child misbehaving in a way that makes your daughter feel she must defend herself?
Jealousy usually fades with time, as long as the children are both getting appropriate amounts of attention, and the other child isn't allowed to act inappropriately toward your daughter. For example, even though the 15mo's desire to play with whatever your daughter is playing with is normal, that does not make it appropriate. Kids learn appropriate behavior only when it is modeled by their role models as consistently as bad behavior must be punished. Neither the younger child's age, nor your daughter's inappropriate behavior may be treated as an excuse for inappropriate behavior on the younger child's part, or the situation won't get better. Rules are rules for everyone, and your daughter won't value rules unless they protect her (say, from the other child taking her toys away) as well as limit her.
Discipline for a 15 month old is, of course, different from that for a 2yo. The 15mo should be given a warning when she misbehaves, and then removed from the play area immediately upon the second infraction. Stick her in a playpen or similar confinement without toys while the adult present gives attention to your daughter for a few minutes (and pretends the 15mo isn't there without regard to any amount of screaming or other protest). Do not take her out until a couple of minutes have passed AND she is quietly behaving. She'll start to learn that bad behavior is no fun.
Both of the children will resist the change at first -- they will probably increase the bad behavior in the hope that it will go back to getting the reaction it used to -- but within a couple of weeks you will see a great improvement in their behavior. However, any inconsistency, or giving attention in time-out, etc. will undo your efforts. Additionally, reacting 5 or even 3 minutes after a bad behavior is too late for a child this young to connect the behavior with the punishment. Your or the nanny's response must be immediate, or the punishment will seem random to her rather than the consequence of a particular behavior.