It is not incorrect to use "in what degree", but the most common way of saying it is "to what degree".
Some statistics:
IN AMERICAN ENGLISH
"TO WHAT DEGREE" occurred 99.01%
"IN WHAT DEGREE" occurred 00.98%
IN BRITISH ENGLISH
"TO WHAT DEGREE" occurred 98.94%
"IN WHAT DEGREE" occurred 01.06%
IN CANADIAN ENGLISH
"TO WHAT DEGREE" occurred 95.95%
"IN WHAT DEGREE" occurred 04.05%
IN THE ENGLISH USED IN IRELAND
"TO WHAT DEGREE" occurred 99.97%
"IN WHAT DEGREE" occurred 00.03%
IN AUSTRALIAN ENGLISH
"TO WHAT DEGREE" occurred 98.46%
"IN WHAT DEGREE" occurred 01.54%
IN NEW ZEALAND ENGLISH
"TO WHAT DEGREE" occurred 99.98%
"IN WHAT DEGREE" occurred 00.02%
IN INDIAN ENGLISH
"TO WHAT DEGREE" occurred 95.65%
"IN WHAT DEGREE" occurred 04.35%
This statistics shows the degree of rarity of "in what degree". No wonder you being an Indian, as that is what I suppose, found that phrase in your Indian book. Interestingly there is no occurrence of "in what degree" in the English used in your neighboring countries, that is Pakistan and Sri Lanka (during the period this statics is based on).