"indeed exist" is correct and usable in written English.
You can use it to emphasize the fact that something is real or present. For example, you might say, "It is clear that these difficulties indeed exist."
But Georgy Catoire does indeed exist, and is worth hearing.
In a return letter, Mr. Fitzpatrick insisted that such tests did, indeed, exist.
However, running out of ideas does indeed exist – and when it happens it can be scary.
That may indeed exist below the democratic patina of these declarations.
The house was bought very early on from the daughter, who did indeed exist.
I can now safely report back that Suriname, splendid in its isolation, does indeed exist.
But Dr. Moran, the Suffolk poet laureate, has argued for a decade that a Long Island school does indeed exist.
So nuclear critics, such as CND's Kate Hudson, are correct to assert that nuclear subsidies do indeed exist.
According to Gilligan, a sound medical basis for Teddy's condition does indeed exist: he could suffer dissociative identity disorder.
If they indeed exist, all these new supersymmetric particles must be heavy to have escaped detection so far.
As it happens, guns that work only when they are in the hands of their legal owners do indeed exist.
