The verse is talking about the positions of the nine planetary deities in the seven Chakras viz - Muladhara (near the anus), Swadhisthan (the Chakra near the navel), Manipura (the seat of the Fire), Anahata (heart Chakra), Vishuddha (throat Chakra), Ajna Chakra (the 3rd eye Chakra) and Sahsrara (the top of the head).
ādhāre prathame sahasrakiraṇaṁ
Adhara=base; Prathama=first.So, adharae prathame= in the first base i.e. in the Muladhara. Sahasrakiraṇaṁ - Thousand-rayed= another name for Sun as noted in Surya Kavacha Stotra by Yajnvalkya. So, Sun is in the root Chakra or Muladhara.
Tara=Stars and Dhavam here means Master or Lord. The Moon god's one name is Tarapati (lord of the stars) as noted in the Chandra Kavacham. Swa refers to the 2nd Chakra the Swadhisthana. Asraya=rests,depends etc. So, Moon belongs to the Swadhisthana.
Maheyam from the word Mahi (meaning Earth) refers to Mangala who is considered as the son of Earth. Mars is situated in the seat of Fire or in the Manipura Chakra.
hṛdi budhaṁ kaṇṭhe ca vācaspatim
Hrudi= in the Heart, Kanthe- in the Throat
In the heart (Anahata) Budha and in the throat (kanthe) Brihaspati (Vacaspati meaning "the Lord of speech" is another name for Brihaspati).
Bhrumadhye=The place between the eyebrows=where the Ajna Chakra is placed. Brigunandanam=son (nandana) of Bhrigu referring to Shukra.
dinamaṇeḥ putraṁ trikūṭasthale
Dinamani is the Sun, so Dinamaneh Putra=Sun's son (putra) referring to Shani. Kutastha in Yoga and Tantra refers to place higher than the Ajna Chakra and near the Sahsrara. It is kind of a "Highest Place".
In the veins reside Rahu and Ketu.
I could not get the meaning of the last part but Gulika is the name of an Upa-Graha.
Regarding the location of the verse, even I don't know that like you but I want to.