forgot to tell / telling her.

Both sentences are fine.

They mean entirely different things - but are both fine.

Thank you, ehBeth. What is the difference?

Can someone else please help since Beth isn't around?

I forgot telling her to write to you.
This is awkward as it stands. We generally use the infinitive with forget (not a gerund) if we wish to convey that we failed to remember to do something. I would prefer "I forgot about telling her to write to you." However it is ambiguous, since it could mean you forgot that you had already told her, or it could mean you forgot to tell her, which brings me neatly to...

I I forgot to tell her to write to you.
... which is what I would write.

Remembering/forgetting gerund and infinitive comparison

I remembered buying a train ticket - I remembered (in the past) a previous occasion on which I bought a train ticket. (Use 'forgot about' for the converse meaning).

I remembered to buy a train ticket - I did not fail (in the past) (due to a lapse of memory) to buy a train ticket which I would need later. (Use 'forgot' for the converse meaning).

Thanks, centrox.

The reason, Tang, you received that mysterious letter from her was that

Otherwise of course it wouldn't have seemed so mysterious

...tho still clear as molasses