I'm impressed by your good intentions.

Your lady will give you the signs for you to approach her. If she smiles at you that is a very good sign. If she is open to talking to you and gives you the impression that she enjoys your company then you are already in-it-to-win-it.

You already don't give the impression that you consider women to be pieces of meat, that already shows an amount of character on your part. Build on this. Tell your lady friend just how much you enjoy her company and just how much you would like to enjoy it again.

You don't need over-the-top means of flattery, just an honest appreciation of the time you spend together is enough for her to pick up on your intentions.

A few days ago she made a bet with me that I could not cook her a satisfactory dinner.

She did not make this bet out of a purely academic interest in your culinary skills. Nor does she expect you to pick something material like restaurant coupons as your reward. She's already interested, or she would not have had a private dinner with you. She's likely expecting you to ask for some kind of future date together as your 'reward'.

Also, I am guessing that English is not your first language, and I'm guessing that she did not call herself 'vanquished'. You vanquish a mighty foe. A woman is not a foe to vanquish.

Challenge her back.

She challenged you, thereby showing an interest in you (she could've just said no; this is a more challenging way to say yes). Now it's your turn. This could keep things interesting between you two.

You could make plans to do something you're good at, and dare her to keep up. Or if you prefer going to the park, challenge her to create a fun picnic.

I assume this is in the context of western culture. I think you should not only ask for her company. You invested some time and your cooking skills. Therefore you can ask of somethink similar. For example let her prepare a picnic, of course you should ask for something which will lead to her spending time with you. The reason I suggest this is, that humans tend to hold on something more likely, if they already invested something (time, money...). This behaviour is called sunken cost bias. If you just ask for her time, she will think that her spending time with you is a reward for you. So every time she will spend time with you she is rewarding you and therefore you have to earn her time.

She made a bet with me that I could cook not her a satisfactory dinner.

The bet isn't about something you can prove, it's not a 1 + 1 = 2. It's subjective. Therefore, "biased", to say the least while still being polite. This way, she can choose to let you win... or not!

And, doing that, gives you a 1st hint. Then, follows this:

Because I won she said that I get to pick a reward.

Was the reward planned? No. Another hint. Usually, when you bet with friends/family/colleagues, depending on the amount of fun you want to have, you choose a reward on the serious-to-silly gauge. But it's known and agreed before.

Here, IMO, she opens a door, and want to see how you'll step inside. Depending on what reward you'll pick, she'll see how serious-to-silly you can be. Will you be friendly? Funny? Goofy? Romantic? Yourself?

Well, she wants to know who you really are, how far you can go. And how you like to go there.

My best advice is to give her what she wants: be yourself, let her know who you are, be honest. You want to spend time in a friendly manner? Just do that.

What seems important, to me, is the way you offer/take the/her reward:

Hey, Alice, as you lost the bet, you'll have to do the worst thing ever! (puzzled..) Yes! Spend a couple of hours with me [ walking in the park and having a coffee / visiting this museum / you-name-it ]. Maybe with humor? :)

Watch her face / smile (hopefully...) and you'll know more. But don't be afraid of the answer.

As I said in the question's comments, I think this is strongly culturally-related, but let's take a shot in the dark:

In many regions of the world, such a bet would already be considered as flirting, so it would not seem awkward to ask for spending some time to get to know her better. With such an hypothesis done, it could go that way:

Hey X, I thought about what I could pick as a reward, and it would really please me to spend some more time together. I think you're an interesting person and I would like to get to know you better.

I think this is quite neutral yet honest sentence, but please keep in mind the cultural background: I believe people in the US are not shy to tell their feelings before even going on the first date, whereas in my country, it's more of a "catch the mouse" game: I won't tell you I like you until I feel you do too, and vice versa.

It's pretty close to impossible to comment on someone else's motives, especially when they aren't here to clarify their position. All we can comment on is your course of action.

But let's look at this. She willingly made a bet with you. She was the judge of whether you won or lost and it's purely subjective. You get to choose the prize. How easy would it have been for her to say "you lose"?

Something I've learned over the years is that we as men don't tell women often enough what we want. They wonder why we don't do [x] and we wonder why we don't get a clear signal so we can do [x]. The solution is to put ourselves out there in a way that makes our intentions known and to try.

What do you want? It appears that the answer is "to spend a couple hours with her, touring the city". So tell her! "What I would like more than anything is the pleasure of your company for a couple hours as we tour the city." Make it clear that it's about her and that you enjoy spending time with her. Given that you spend time together already, I can't see that she would find this as excessive. But I'm not a mind reader, so it's impossible to say she'd say yes.

I'd say this: she's 36. She's at a point in her life where she's realized that getting what she wants is up to her. She's most likely gone through dating. She most likely doesn't want to play games and doesn't like the hesitancy of guessing games as part of dating. The only way you'd know if she doesn't like this, however, is for you to ask. So do that! Or spend the rest of your life wondering "what would have happened if..."