How do I properly set the Datetimeindex for a Pandas datetime object in a dataframe?

To simplify Kirubaharan's answer a bit:

df['Datetime'] = pd.to_datetime(df['date'] + ' ' + df['time'])
df = df.set_index('Datetime')

And to get rid of unwanted columns (as OP did but did not specify per se in the question):

df = df.drop(['date','time'], axis=1)

You are not creating datetime index properly,

format = '%Y-%m-%d %H:%M:%S'
df['Datetime'] = pd.to_datetime(df['date'] + ' ' + df['time'], format=format)
df = df.set_index(pd.DatetimeIndex(df['Datetime']))