I am not a native speaker, so please take this with a grain of salt. I only know classical Arabic. Modern usage may have pitfalls...
As to the best of my knowledge ليس لدى is just a more emphatic version of ليس لى. As Arabic doesn't belong to those languages having a word for "to have" like e.g. Polish, German, English, or French there is another way of indicating possession, like in Latin (mihi est ...) or in Russian (у меня ...). The Arabic "solution" for "to have" works exactly like Latin. So the verb involved ليس (not to be) is in a neutral form (3.Pers.Sg.) and does not agree with the possessor.
I hope this helps.
transcription of the above, using classical i'ráb: anâ faqîrun jiddan, wa laisat ladayya ayyata muahalâtin 'ilmiyyatin.
The grammatical subject is "ayya" which due to the rule of "laisa wa akhawâtuhâ" is in accusative. And because "ayya" is feminine, the verb must be 3.Pers.Sg feminine i.e. "laisat".
You Arabic native speakers out there, reading this, please correct and complete! من فضلکم