"पुरा येनेक्षिता जानकी" has only passive past participle in feminine gender from
ईक्ष् īkṣ [ IkS ]
cl. 1. Ā. ī'kṣate, īkṣāṃ-cakre ( Pāṇ. i, 3, 63 ), īkṣiṣyate, aikṣiṣṭa, īkṣitum, to see, look, view , behold, look at, gaze at
to watch over ( with acc. or rarely loc. ), AV. AitBr. ŚBr. Mn. Kathās. etc.
to see in one's mind, think, have a thought, ŚBr. ChUp. MBh. Bhag. etc.
to regard, consider, Kum.
to observe ( the stars etc. ), VarBṛS.
to foretell for ( dat.; lit. to observe the stars for any one ) Pāṇ. i, 4, 39 : Caus. īkṣayati, to make one look at ( with acc. ), ĀśvGṛ. ( This root is perhaps connected with akṣi, q.v. )
http://www.sanskrita.org/wiki/index.php/IkS
in other cases, it can be from the same root it can be active present participle in masculine singular nominative:
ईक्षिता
वि० [ईक्षितृ] देखनेवाला [को०] ।
īkṣitṛ [ IkSitR ]
mfn. seeing, beholding, a beholder, Mn. Prab. etc.
So without context it cannot be explained any way. If he had asked, either of the forms ईक्षित or ईक्षितृ it would have been clearly be derived from the same root to get the desired meaning.
The same with ईप्सिता ---
from the same desiderative verb ईप्स्
irrespective of Sanskrit to English or Kannada or any language to which it is to be translated:
( Desid. of √ āp, q.v. ), to wish to obtain.
nominative singular in masculine from ईप्सितृ (pr.p) or from in feminine from ईप्सित (past participle)