There's noting wrong with your lime that plenty of good, all-round
sunlight won't cure. These plants need intense light and the partial
etiolation of the stems is indicative of one that has been kept as a
house plant. Ideally, it should be placed in a warm, wind-sheltered,
sunny spot out of doors as soon as warm weather arrives in late spring/
early summer and allowed to remain out until nights start to drop
below 5C - usually from September in the north to October/early
November in the far south,. It does not need potting on at the moment
and can remain undisturbed until next spring at the earliest. Yours
looks to be in a 15cm. diameter pot at the moment and it can be moved
to a 20cm. pot next year. They seem to do best if the roots are
slightly cramped.

You can buy ready-made composts suitable for Citrus, but I find they
need a small amount of additional coarse horticultural sand or grit
added to give them the porosity that this and allied Citrus prefer. I
prefer to make my own using a mixture of a good quality ericaceous
compost, good John Innes No 2 (loam based) and additional fine grit an
a ratio of 4:4:1. A couple of drenches of diluted magnesium sulphate
(two heaped tablespoons per gallon) applied at 3 week intervals from
early March will ensure deep green healthy growth followed by Chempak
No2 at the recommended dilution rate every 3 weeks during summer.