There are several reasons why it's easier for your wife to do it. Some are obvious -- I'll skip those -- but one that might not be so obvious is that your wife's approach is more cut and dried.
Here are some suggestions (I assume that's what you're really after...).
Have a family meeting (the three of you). In the family meeting, you will tell your daughter that you are going to institute a monthly special father-daughter outing. And you will ask of her that she be less clingy (you'll have to figure out how to explain this concept to her -- a bit of role play might help). Give her a simple I-message about how you feel when she is clingy, but also tell her how much you enjoy spending special time with her. Do some brainstorming together to find out what sorts of special outings would appeal to her.
(You are probably doing this already --) Help your daughter feel that she has some important responsibilities with the baby, for example, explaining things to the baby. Find lots of opportunities to remark to your wife (in your daughter's hearing) how the baby perked up when your daughter walked in the room, how interested the baby is in its sister, etc.
Plan out a bedtime routine for your daughter containing some rituals. You can gradually help her transition to expressing some of the lovey-dovey stuff with a favorite stuffed animal instead of all with you.
When it's time to turn out the light, you have two options.
(a) As you are getting her settled, give her a verbal preview of what's to come, that when she's all ready for bed, and you've finished reading a picture book, you will kiss her good night, turn out the main light, and go wash the dishes; and then you will check on her every five minutes or so. It's okay if she cries a bit. In that case, you just bend time a bit, and make the "five minutes" go by very quickly. You will gradually be training her that you haven't forgotten about her, and you always come back. Start this procedure on an evening when it won't be a disaster if she's short on sleep the next day. It's very important that you not cave in and ask your wife to go in. Your wife should be out of the house, asleep, or otherwise occupied!
When you go back in, ask if she's too warm, or too cold, or if anything hurts or itches; get her comforter nicely arranged again, check if she has the right stuffed animal. Your voice should sound eminently reassuring. You can utter a formula if you like (but only once per visit), such as, "Mmm, it's so nice to be in my warm, comfy bed with [name of stuffed animal]."
(b) Get comfortable in a moveable chair in her room, with a nice podcast to listen to, while she's falling asleep. You'll have to experiment to find the optimal distance from her bed for your chair. Your presence should be reassuring, but you don't want to be so close that the stimulation of having you there keeps her from falling asleep.
The podcast will make it easier for you to not engage with her. "I'm going to listen to my program now, we'll talk about that tomorrow. And if necessary, broken record "We'll talk about that tomorrow.... Tomorrow."
If you start to get sleepy and are afraid you are about to fall asleep, and she's still awake and will be alarmed if you leave, say, "I'm going to the bathroom, I'll be right back." Now, interestingly, this may well be the point at which she drops off -- while you're out of the room!
Let me know if you have any questions!