Who discovered DNase?
There are, of course, many different deoxyribonucleases (DNase), but the question has other implicit problems. It assumes that there was a single event that resulted in the realization of the existence of a new enzymatic activity. As the Introduction to this paper by M. McCarty on the purification of one such enzyme in the Journal of General Physiology in 1946, science wasn’t (and still often isn’t) quite like that:

The enzyme which breaks down highly polymerized desoxyribonucleic (sic) acid
has been known for many years under various names. In 1903, Araki (3)
reported that crude preparations of trypsin and erepsin brought about
liquefaction of gels of a thymonucleate prepared according to the
method of Neumann (4) Although Araki interpreted the effect as being
proteolytic in nature, his experiments represent the first observations
on the enzymatic breakdown of this type of nucleic acid. Two years
later Sachs (5) demonstrated that liquefaction of the nucleate gel by
pancreatic extracts is caused by an enzyme distinct from trypsin, and
further presented evidence indicating that trypsin rapidly inactivates
the nuclease. De la Blanchardière (6) confirmed the findings of Sachs,
and made several unsuccessful attempts to separate the nuclease from
the inactivating tryptic enzymes. This writer also devised a more
quantitative method of measuring the nuclease activity by following the
fall in viscosity of a dilute solution of thymonucleate during
digestion with the enzyme.

Araki, T., Z. physiol. Chem.,1903,38, 84.

Sachs, F., Z. physiol. Chem.,1905,46, 337.

De la Blanchardière, P., Z. physiol. Chem., 1913, 87, 291



[desoxyribonucleic acid and thymonucleate are older terms for DNA and its ionized form]
Many scientists have subsequent been involved in identifying different types of DNase and their roles in DNA metabolism, and anyone wishing to know about a particular DNase would need to research the literature.