I used to run the DofE for a youth group a while ago so I have fairly detailed knowledge...
There are three stages - Bronze, Silver and Gold.
For all three you have to undertake some form of voluntary work over a period of 3 months for Bronze, 6 months for Silver and 12 months for Gold. Costs for that should be minimal or non-existent.
You also have to take part in physical activities. Depending on which activity you choose, you may simply have to participate or you may have to achieve some standards. The costs for this depend on what physical activities are chosen.
The skills section is required at all three stages. You have to learn a new skill. Again, costs depend on the skill chosen. If your daughter is already learning a musical instrument, she may be able to use that.
You have to participate in the physical and skills activities for 3 months for the Bronze. For Silver you have to do one for 6 months and the other for 3 months - you can choose which way round. Similarly for Gold it is one for 12 months and the other for 6 months.
As others have said, the expedition is often the biggest cost. The Bronze expedition is 2 days and 1 night rising to 4 days and 3 nights for Gold. You have to do a practise expedition as well as the qualifying expedition. The practise doesn't have to be as long as the full expedition but for Silver and Gold it must include 2 consecutive nights away. The expedition can be on foot (the most common choice), by canoe/kayak, boat, bicycle, wheelchair or on horseback. Whoever is operating the DofE scheme should be able to provide some equipment (tents, etc.) but you will have to provide personal equipment such as boots, suitable clothing, rucksack, etc.
Finally, for the Gold award there is a residential requirement. That involves spending 5 days and 4 nights away from home on a shared activity with strangers. Costs vary hugely depending on the choice made. Helping on a children's camp, for example, is likely to cost a lot less than undertaking a project in India.
The Bronze award takes a minimum of 3 months, the Silver needs 6 months and the Gold at least 12 months. Most people take longer than this.
You will certainly be involved in providing a taxi service! It is unlikely you will be involved much more than that - part of the objective is to get young people doing things for themselves.