Word meaning "difficult to eliminate"?

You could say: "I went to great efforts to fix a stubborn bug in my code."

Google defines stubborn as:

difficult to move, remove, or cure.

Tenacious:

1. holding fast; characterized by keeping a firm hold (often followed by of):

   • a tenacious grip on my arm; tenacious of old habits.

2. highly retentive:

   • a tenacious memory.

3. pertinacious, persistent, stubborn, or obstinate.

"I went to great efforts to fix a tenacious bug in my code."

Obstinate is a good word for this, as it describes something that is undesirable and persistent. From Dictionary.com:

1. firmly or stubbornly adhering to one's purpose, opinion, etc.; not yielding to argument, persuasion, or entreaty.
2. characterized by inflexible persistence or an unyielding attitude; inflexibly persisted in or carried out: obstinate advocacy of high tariffs.
3. not easily controlled or overcome: the obstinate growth of weeds.
4. not yielding readily to treatment, as a disease.

If you want a $2 metaphor, you can try Obdurate (Dictionary.com):

1. unmoved by persuasion, pity, or tender feelings; stubborn; unyielding.
2. stubbornly resistant to moral influence; persistently impenitent: an obdurate sinner.

Intractable is my favourite for this sort of thing; not hard, but annoying and difficult to get to grips with.

Resilient would be appropriate.

From Merriam-Webster:

:  characterized or marked by resilience: as
a :  capable of withstanding shock without permanent deformation or rupture
b :  tending to recover from or adjust easily to misfortune or change

Definition 'b' seems especially apt.

In science and medicine literature people often talk about bacteria (something undesirable, like a computer bug) being resilient to efforts used to try and eradicate it.