Beware; there are two Henri IV hotels in central Paris, and not far apart. Or maybe three.
Hotel Residence Henry IV is on the left bank, 50 rue des Bernardins, Paris 75005. http://www.residencehenri4.com/en/
Hotel Henri IV is also in the heart of the Left Bank tourism crawl, 9-11 Rue saint Jacques, Paris 75005 http://www.henri-paris-hotel.com/en/#/location-hotel-paris-saint-michel
Hotel Henri IV, 25, place Dauphine, Paris 75001. Place Dauphne is an obscure, pretty square on Isle de la cite. The place's narrow entrance is across from the very kingly statue of Henry riding his horse. In my penny-pinching youth, a long time ago, the hotel was about as basic as you can. Booking by telephone only. The booking site Eurocheapo says the property was sold at least a year ago but not closed right away. Trip Advisor has at least one review from this spring but, given the possibility of confusion, it appears the reviewer was talking about one of the other hotels.