The two other answers have been rather unwelcoming to India (one of them now deleted), and not just as an Indian, but as someone completely sold into the fact that travelling in India is extremely interesting and fun I would like to answer with a positive bias, strictly speaking I have no knowledge of the other two destinations so I am not competent to compare.
Delhi as a Hub
Your choice is spot on, without much travel (travel less than 500 kms) you have many destinations which can be a treat for a traveller, commutes by Air-conditioned Trains /Cars would make it even better. Agra, Jaipur, Kashmir Valley (though a bit far and if you read this and is convinced about safety) and Delhi itself would be a great experience. As everything around this place would be different, I guess it would keep your teenager engaged and interested in most things here.
One big warning in India, an Australian friend of mine who has travelled multiple times into India clearly identified that eating non-vegetarian food in India can be unsettling, now inferring that the entire country of India is dodgy because you did not understand this fact would be unfortunate, but at times can happen because of your diet habits. I have seen him safely travel through India eating vegetarian food and have no problems at all and have trouble within a couple of hours of eating non-vegetarian, it might just be him, but this is my personal experience. I mention this also because a few friends of mine who have travelled to Brazil and wanted to keep to a vegetarian menu had trouble and so this could be a factor for you.
I believe as a civilisation used to using water in more ways than in the West, seeing water in some place won't make us feel it's unhygienic in itself.
We have wet toilets, and cleaning floors with water is the most common method and so it may strike you as a major problem, and yes there are diseases that spread through water and hence you need to be careful, but if you understand the fact that we use water differently from the cultures which probably did not have sufficient water until about 500 years ago, you may be able to navigate your travel better.
Regarding climate, strictly speaking I Googled up the weather in your place (the location on your profile) and looks like 38 degree centigrade won't be something you have never seen in your life, so 40 or 42 may not be as unfriendly to you as to an average European.
The language aspect, I would say you will probably never find another destination where there is such diversity in language, while English and Hindi are widely spoken, if you travel enough within India, you will feel the diversity of the languages spoken and that could be deeply enriching for a teenager. While with English you can easily survive, you won't find the entirety of India to be totally English friendly.
I can be confident here, India would be the cheapest. Travel and accommodation would be very inexpensive here in India, even if you avoid the bottom 40% as being unsafe/undesirable. Fortunately there are enough review sites on the web and TripAdvisor ratings have actually even become an offline evidence in some hotels, just avoid the dodgy agents who would almost always overcharge a foreigner and book on the web after considering the reviews and you should be fine.
There is an interesting angle about poverty, an IT employee in India can get paid about $5 a day and live a comfortable life in many parts of India. Yes, I did not make a mistake about that number; at $10 a day, families live very comfortably here, the fact that you will pay income tax for anything over $11 a day is evidence to this. This is the reason things are cheap here, so the flip-side to low cost will always be local poverty (or seeming poverty).
Though there are no marked cultural festivals around the time you visit India, you will certainly see a stark difference in the culture here, some of our people still wear traditional clothes and go about their lives so very differently from the western world, I wouldn't be able to compare it to the other destinations here.
Personally if you ask me though, I would agree with JoErNanO,
Since the objective here is to please her, I would let her choose the destination. ;)