I enjoy discover historic places in Ha Noi, and neither does she.
Correction: I enjoy discovering historic places in Ha Noi, and neither does she.
(The verb after "enjoy" needs to be in the -ing form.)

She is at the same old as me.
Correction: She is at the same age as me.
("Old" should be replaced with age because we use "same age" to refer to people being the same in terms of their years.)

He just bought a instrument for his mum on her birthday.
Correction: He just bought an instrument for his mum on her birthday.
("A" should be replaced with an because "instrument" begins with a vowel sound.)

That smells very like to burning.
Correction: That smells very much like burning.
("Very like to" is incorrect; we use "much like" to compare smells.)

How much sandwiches do you want to buy? - Just two.
Correction: How many sandwiches do you want to buy? - Just two.
("Much" is used with uncountable nouns, and "sandwiches" is countable, so we use many.)