The invasion of chilli thrip, which can transmit a host of plant viruses, forced Kew to close the building every Tuesday afternoon until the beast is blitzed.
"There is currently no end date to the closure as this will depend on the success in eradicating the pest," said a spokeswoman.
"Chilli thrip, Scirtothrips dorsalis, is a tiny insect that can transmit plant viruses, some of which may have economic or environmental implications."
Treatment would include removing host plants, pruning out, cleaning infested areas and encouraging biocontrols across the house, careful feed and watering regimes.
She said Kew weighed up the possible impact of chemicals on non-target organisms, but the royal garden had a statutory duty to protect its plant collections.
"Chemicals are used as a last resort and, where possible, only products that specifically affect the target pest."
Chilli thrips attack vegetables as well as ornamental plants and have caused damage in south Asia, the Caribbean and north America.
"The pest was discovered in January 2008 during routine monitoring. We can't be sure where it originated; maybe on flowers used for events or tropical fruits."
She said the thrip was restricted to the palm house, but such a complex house and dense plant canopy made it hard to tackle.