tagelang
for days
Continuing over a period of several days.
Wegen des Schneesturms waren wir tagelang eingeschneit.
Because of the snowstorm, we were snowed in for days.

for days
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for days
Continuing over a period of several days.
Wegen des Schneesturms waren wir tagelang eingeschneit.
Because of the snowstorm, we were snowed in for days.

The word is simply built from 'Tage' (days, plural of 'der Tag') and 'lang' (long). It literally means 'over the length of days' and expresses a duration.
Die Reparatur des Autos dauerte tagelang.
The car repair took days.
As an adverb, 'tagelang' (for days) can appear in different positions in the sentence. It often stands at the beginning of the sentence for emphasis or after the conjugated verb in the middle field.
Tagelang hat es ununterbrochen geregnet.
For days, it rained nonstop.
'Tagelang' (for days) follows a useful pattern: [time unit in the plural] + '-lang'. You can also form other time adverbs this way, for example 'wochenlang' (for weeks), 'monatelang' (for months), 'jahrelang' (for years), or 'stundenlang' (for hours).
Er hat wochenlang für die Prüfung gelernt.
He studied for the exam for weeks.
daily
'täglich' and 'tagelang' both contain 'Tag' (day), but they do not mean the same thing. 'Täglich' describes regular repetition: something happens every day. 'Tagelang' describes a duration: something lasts for several days without an important stop, or continues over several days. The words can be confused because both can talk about several days. If you want to say how often something happens, use 'täglich'. If you want to say how long something lasts, use 'tagelang'.

Wegen des Sturms warten die Reisenden tagelang auf ihren Zug.
Because of the storm, the travelers wait for their train for days.
Der Verkäufer öffnet den Laden täglich um neun Uhr.
The shop assistant opens the store daily at nine o'clock.